M Riou

F Abreu Araujo

J Torrejon

S Tsunegi

G Khalsa

D Querlioz

P Bortolotti

V Cros

K Yakushiji

A Fukushima

H Kubota

S Yuasa

M D Stiles

J Grollier
email: julie.grollier@cnrs-thales.fr

Neuromorphic Computing through Time-Multiplexing with a Spin-Torque Nano-Oscillator

Neuromorphic computing through time-multiplexing

with a spin-torque nano-oscillator

I. INTRODUCTION

Reservoir computing is a neural network-based theory, designed to process temporal inputs (Fig. 1) [START_REF] Maass | Real-Time Computing Without Stable States: A New Framework for Neural Computation Based on Perturbations[END_REF], [START_REF] Jaeger | Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in Wireless Communication[END_REF]. A reservoir is composed of non-linear units, or neurons, that are connected recurrently through fixed connections (Fig. 1a). Input waveforms modify the activities of neurons inside the reservoir and the perturbed activities propagate through the recurrent network. The outputs of the reservoir are linear combinations of some or all neuron responses in the reservoir. The coefficients of these weighted sums are trained to obtain the desired outputs. A reservoir can classify waveforms due to the non-linearity of its neurons. It can also perform prediction tasks due to the recurrent connections which allow for fading memory of past inputs [START_REF] Appeltant | Information processing using a single dynamical node as complex system[END_REF]. The limited number of connections to train make reservoir computing an excellent approach to evaluate novel technologies for neuromorphic computing.

In particular, we have recently shown experimentally that a single nanodevice, the spin-torque nano-oscillator, can implement reservoir computing through time multiplexing [START_REF] Torrejon | Neuromorphic computing with nanoscale spintronic oscillators[END_REF]. Spin-torque nano-oscillators (Fig. 2a) are magnetic tunnel junctions driven by dc current injection into a regime of sustained magnetization precession [START_REF] Grollier | Spintronic Nanodevices for Bioinspired Computing[END_REF]. Magnetic oscillations are converted into voltage oscillations through tunneling magneto-resistance (Fig. 2c). Despite the sub-micrometer diameter of such a spintronic artificial neuron, the experimental results for spoken digit recognition reach the state of the art for existing hardware and software (99.6 % recognition). In the present work, by analyzing a simple task, the classification of sine and square waveforms, we give a recipe for high classification performance with spin-torque nano-oscillators.

II. EXPERIMENTAL PROCEDURE

Our oscillators have FeB free layers with diameters of 375 nm, and a magnetic vortex as a ground state. A schematic of the experimental set-up is provided in Fig. 2b. We use the dc current to set the amplitude of voltage oscillations in the absence of inputs, and apply the input waveforms as a superimposed ac current. To compute, we use the fact that the amplitude of voltage oscillations across the junctions, , is a non-linear function of the injected current ( Fig. 2d).

Using time-multiplexing, a reservoir can be emulated with a single oscillator, which plays the role of all neurons one after the other (Fig. 1b) [START_REF] Appeltant | Information processing using a single dynamical node as complex system[END_REF]. This strategy requires that the state of this neuron at time t+dt depends on its state at time t, just as a downstream neuron usually depends on the state of upstream neurons. This behavior can be achieved by pre-processing the input through multiplication with a binary fast-paced sequence that drives the oscillator into a transient state.

Figure 3 illustrates the procedure for reservoir computing. An input waveform, composed of randomly arranged sine and square waves with the same period, is shown in Fig. 3a [START_REF] Paquot | Optoelectronic Reservoir Computing[END_REF]. The pre-processed input is displayed in Fig. 3b. It multiplies the input segment-wise with a binary sequence which has a total duration and is composed of points separated by a time interval ( = / ). The number of points in the binary sequence, , defines the size of the emulated network. For clarity, we have chosen in Fig. 3 to illustrate the working principle with a small neural network of only 12 neurons ( = 12). The non-linear response of the amplitude of voltage oscillations to the pre-processed input is shown in Fig. 3c.

The response of the network is shown in Fig. 4a in a zoom-in of the response to a single input segment of duration . Due to the oscillator non-linearity, each voltage amplitude value (i = 1…12) is a non-linear transform of the input value. In addition, depends on . Indeed, magnetic oscillations have a relaxation time of about 300 ns, larger than the time interval = 100 ns between each . The equivalent ring neural network is illustrated in Fig. 4c. The output of the neural network (Fig. 4b) is a sum of all : output = ∑ weighted by the strength of each connection, chosen to make the output match the target. The target for the output is a constant value for each waveform: one for squares, and zero for sines [START_REF] Paquot | Optoelectronic Reservoir Computing[END_REF]. This output is reconstructed on a computer from the sampled experimental oscillator response by calculating the optimal weights through matrix inversion. Fig. 5 shows the reconstructed output obtained by experimentally emulating a 24-neuron network. The root mean square (rms) deviation between target and output is 11 %, which is small enough to distinguish between sines and squares without any error (perfect classification) for the chosen choice of parameters: dc current = 7.2 mA, magnetic field = 447 mT, input amplitude = 500 mV (equivalent to 6 mA).

III. RESULTS

The classification performances vary strongly depending on the experimental parameters. Indeed, the voltage amplitude (Fig. 6), the non-linearity ⁄ (Fig. 7) and the voltage noise ∆ (Fig. 8), vary considerably with dc current and magnetic field. Since spin-torque oscillators have a small magnetic volume, thermal noise affects the magnetization dynamics. The resulting voltage amplitude noise is large for large non-linearity, which quantifies the sensitivity of the system to perturbations [START_REF] Slavin | Nonlinear Auto-Oscillator Theory of Microwave Generation by Spin-Polarized Current[END_REF]. The correlation between voltage noise and non-linearity appears clearly in the comparison of Figs. 7 and8. Neuron non-linearity is a key ingredient for classification as it allows the separation of input data [START_REF] Appeltant | Information processing using a single dynamical node as complex system[END_REF]. On the other hand, noise in neuron response is detrimental for classification as it directly affects the output ( ∑ ).

Fig. 9 shows the classification performance as a function of dc current and field. We find good performance by choosing a bias point with intermediate non-linearity and therefore intermediate noise, and where the neuron output changes strongly in response to the ac input. Such bias points allow enough non-linearity to classify while keeping large enough signal to noise ratios to distinguish between outputs. As can be seen from Fig. 10, the larger the ac input variations (between 300 mV and 500 mV), the lower the rms deviations between output and target. Indeed, larger inputs lead to larger responses and improved signal to noise ratios. Fig. 10 shows the evolution of the rms deviations between the output and the target as a function of the length of the time interval between samples. The evolution is completely different when the target is in exact phase with the input (Fig. 10a) and when the target is shifted by /2 with respect to the input (Fig. 10b). Indeed, classification of sines and squares requires the network to have a short term memory of past inputs: some input points in the two different patterns are identical (+1 and -1 at their extrema). Therefore they can only be distinguished if the output of the network depends on the previous values of the input. When input and target are in phase (Fig. 10a), the only source of memory in the network comes from the relaxation time of the oscillator, of the order of 300 ns in our case [START_REF] Appeltant | Information processing using a single dynamical node as complex system[END_REF]. This is why the classification performances degrade for > 300 ns in Fig. 10a. For sampling intervals that are too long, does not depend on anymore, and input sines and squares become difficult to separate. When is much smaller than the oscillator relaxation time, the oscillator cannot respond to the rapidly varying pre-processed input. Changes in become very small, the signal to noise ratio degrades and poor classification follows.

There is an optimum for the sampling interval (in our case = 100 ns for ac input amplitudes of 500 mV). However, there is another way to endow the network with memory. Since the output is reconstructed offline after recording the whole response to inputs, it is possible to shift the target with respect to the input [START_REF] Larger | High-Speed Photonic Reservoir Computing Using a Time-Delay-Based Architecture: Million Words per Second Classification[END_REF]. In that case, some of the samples used for reconstructing the output belong to the previous segment . In other words, the current output is reconstructed partly from the present value of the input and partly from the last value of the input. The best results with this strategy, shown in Fig. 10b, are obtained when the number of samples is evenly distributed between the past and current input value. As in Fig. 10a, results are bad for small due to the low signal to noise ratio but they do not degrade for large values. The artificially introduced memory compensates for the loss of intrinsic memory.

IV. CONCLUSION

Spin-torque nano-oscillators naturally provide the important features for implementing a reservoir computer. The recipe for high performance classification is the following: intermediate non-linearity and a high signal to noise ratio in the neural output. For tasks requiring short term memory, the intrinsic memory coming from magnetic relaxation times can be sufficient. When output reconstruction is done offline, an alternative strategy to endow the network with longer term memory is to shift the target with respect to the input. In the future, it will be interesting to introduce on-line long term memory through time-delayed feedback strategies.

Fig. 3 .

 3 Fig. 3. (a) Input waveform. The task is to discriminate sines from squares at each red point. There are 8 discrete red points in each sine and square waveform. (b) Zoom-in on the preprocessed input waveform for a sine and a square. The corresponding fast binary input sequences are numbered from 1 to 16 (8 for the sine, 8 for the square. (c) Envelope of the experimental oscillator emitted voltage amplitude (µ0H = 466 mT, IDC = 7 mA). The trajectories created in response to the input waveform are numbered from 1 to 16 (8 for the sine, 8 for the square).

Fig. 4 .Fig. 5 .

 45 Fig. 4. (a) Oscillator voltage amplitude changes corresponding to a single time segment τ: Here, 12 neurons (12 samples separated by the time step θ) are used to construct the output. (b) The transient states of the oscillator give rise to a chain reaction emulating the neural network with a ring structure (c) Target for the output reconstructed from the voltages in each time segment τ: output ∑ . Fig. 5. Reconstructed output (red) and target (grey) in response to an input waveform with 80 randomly arranged sines and squares. The magnetic field is µ0H = 447 mT, and the applied current 7.2 mA. The results are based on 24 neurons separated by θ = 100 ns.

TFig. 1 .

 1 Fig. 1. (a) Illustration of reservoir computing concept. The network is composed by a large number of interconnected non-linear neurons. The internal connections are kept random and fixed and only external connections are trained. (b) Single neuron reservoir computing approach using time-multiplexing: the input is preprocessed in order to emulate neurons interconnected through time.Fig. 2. (a) Spin-torque nano-oscillators used for neuromorphic computing: magnetic tunnel junction (CoFe/MgO/FeB) driven by spin transfer torque. (b) Schematic of the experimental set-up. A dc current IDC as well as a fast-varying waveform encoding the input Iin are injected in the spin-torque nano-oscillator. (c) The microwave voltage emitted by the oscillator is measured with an oscilloscope. For computing, the amplitude of the oscillator is used, and measured directly with a microwave diode. (d) Voltage amplitude as a function of current at µ0H = 430 mT. The typical resulting excursion of current amplitude is highlighted in magenta when an input signal with maximum amplitude ±3 mA (corresponding to ±250 mV), is injected.

Fig. 6 .

 6 Fig. 6. Amplitude Voltage of the oscillator in the steady state: map in the IDC -µ0H plane.

Fig. 9 .

 9 Fig.9. Root mean square of output-to-target deviations: map as a function of dc current IDC and magnetic field µ0H. The oscillator voltage amplitude curves (in red) in response to the input waveform (in gray) are plotted for selected dc currents (3, 4.5, 6.5, 7.5 and 9) mA and magnetic field µ0H = 380 mT. Here Vin= 300 mV and θ = 100 ns is used. RMS map corresponds to the target shifted by /2 with respect to the input.

Fig. 10 .

 10 Fig. 10. Root mean square of output-to-target deviations as a function of the time step θ (separation between transient states of the oscillator ) for different amplitudes of the input signal (300, 400 and 500) mV: (a) the target is in exact phase with the input and (b) the target is shifted by /2 with respect to the input.

Fig. 7 .

 7 Fig. 7. The non-linearity ⁄ of the oscillator: map in the IDC -µ0H plane.

Fig. 8 .

 8 Fig. 8. Amplitude noise ∆ of the oscillator in the steady state: map in the IDC -µ0H plane.

ACKNOWLEDGMENT

This work was supported by the European Research Council ERC under Grant bioSPINspired 682955.


Damir Vodenicarevic

Alice Mizrahi

Nicolas Locatelli

Julie Grollier

Damien Querlioz
email: damien.querlioz@u-psud.fr

Spintronic Nanoscillators for Unconventional Circuits

Spintronic nanodevices -which exploit both the magnetic and electric properties of electrons -are a major breakthrough for nanoelectronics. The flagship device of spintronics, the magnetic tunnel junction, naturally provides a binary memristive device with outstanding endurance, high speed and low energy consumption. Interestingly, relying on the same physics as their use as memristive devices, magnetic tunnel junctions may also be used as a type of voltage controlled oscillators: at a given voltage, oscillations in their electrical resistance and currents can be observed. Depending on the tuning of the devices geometrical and magnetic properties, the oscillations can be slow and stochastic, or fast and quasi-harmonic. In this work, we introduce experimentally validated models for these two regimes. We discuss the synchronization potential of these oscillators, as well as their possible use in unconventional circuits. Prospects for bioinspired systems are given. This work raises important questions regarding the use of nanooscillators in circuits, which we discuss in conclusion.

I. INTRODUCTION

Spin electronics, or spintronics, is a major breakthrough of micro-and nano-electronics, highlighted by the Nobel price awarded to Albert Fert and Peter Grünberg in 2007. A spintronic device function is using not only the charge, but also the intrinsic magnetic moment, or spin, of electrons. The flagship device of spintronics is the spin torque magnetic tunnel junction, which can be viewed as a binary memristive device, with multiple advantages: it is compact and scalable, its endurance is outstanding, switching is fast and low-energy [START_REF] Diao | Spin-transfer torque switching in magnetic tunnel junctions and spin-transfer torque random access memory[END_REF]. These properties make this device an appealing candidate for standalone and embedded non-volatile memory applications, and multiple industries are now developing spin torque magnetic random access memory (ST-MRAM) exploiting them. A significant drawback is the stochastic nature of switching of these devices [START_REF] Diao | Spin-transfer torque switching in magnetic tunnel junctions and spin-transfer torque random access memory[END_REF], which needs to be addressed through adapted circuit techniques. These stochastic effects could also be exploited, reinterpreting spin torque magnetic tunnel junctions as artificial synapses [START_REF] Vincent | Spin-Transfer Torque Magnetic Memory as a Stochastic Memristive Synapse for Neuromorphic Systems[END_REF], [START_REF] Querlioz | Bioinspired Programming of Memory Devices for Implementing an Inference Engine[END_REF].

However, and quite strikingly, through the exploitation of the same physics as their use for memristive devices, spin torque magnetic tunnel junctions can also behave as oscillators [START_REF] Dussaux | Large amplitude spin torque vortex oscillations at zero external field using a perpendicular spin polarizer[END_REF], [START_REF] Locatelli | Noise-Enhanced Synchronization of Stochastic Magnetic Oscillators[END_REF]. These oscillations are voltage-controlled and are exhibited in the electrical resistance of the device, and observed in two different regimes: slow stochastic oscillations, or fast (microwave) deterministic oscillations. The properties of the oscillators can be designed through materials properties and device sizing, as well as through a magnetic field. In this work, we review these two regimes and propose experimentally validated models that can be used for circuit theory. We discuss to what extent these oscillators can be controlled, as well as their synchronization properties. We propose examples of prospects for bioinspired circuits.

II. STOCHASTIC NANOOSCILLATORS BASED ON SUPERPARAMAGNETIC TUNNEL JUNCTIONS

Spin torque magnetic tunnel junctions are composed of several layers of materials: two magnetic layers separated by an oxide layer (Fig. 1(a)). One of the magnetic layer has fixed magnetization, while the other layer, the free layer, can have its magnetization switched parallel or anti-parallel to the fixed layer. These orientations, P and AP, constitute the two stable states of the magnetic tunnel junction and correspond to different electrical resistance due to the tunnel magnetoresistance effect. By applying a negative or a positive current to the junction, through the physical spin transfer torque effect, the state of the junction can be switched (Fig. 1(b)). This effectively makes spin torque magnetic tunnel junctions binary memristive devices.

The stability of the two states of spin torque magnetic tunnel junctions used as memristive devices is determined by their sizing and their materials properties [START_REF] Diao | Spin-transfer torque switching in magnetic tunnel junctions and spin-transfer torque random access memory[END_REF]. When used as non-volatile memories, those properties are chosen for high stability (i.e. retention higher than 10 years). However, it is also possible to intentionally design spin torque magnetic tunnel junctions for low stability. In this case, the junction is called "superparamagnetic" and naturally switches between its two states due to thermal noise, following a Poisson process (Fig. 2) [START_REF] Locatelli | Noise-Enhanced Synchronization of Stochastic Magnetic Oscillators[END_REF]- [START_REF] Mizrahi | Controlling the phase locking of stochastic magnetic bits for ultra-low power computation[END_REF]. The "natural frequency" of such junctions, corresponding to their mean number of oscillations per unit of time, is:

F = 1 < τ AP > + < τ P > , (1)
where < τ AP/P > are the mean dwell times in the P and AP states. As switching is thermal, it is slow. In Fig. 2, the device switches at kHz rates. It can go in the range of MHz frequencies.

The device behavior can effectively be modeled using a Arrhenius-Neel two-state analysis, meaning that the device switches following a Poisson process. The switching rate is 1/< τ AP > in AP state and 1/< τ P > in P state. In other words, if the switching rate is constant, the probability to switch between times 0 and t is given by

P AP/P (t) = 1 -exp(- t < τ AP/P > ), (2)
The mean times < τ AP/P > are given by: < τ AP/P >= τ 0 exp( ∆E k B T ) where ∆E is an energy barrier separating the two states of the junction, τ 0 is called effective attempt time, T is the temperature, and k B the Boltzmann constant.

The relative stability of the two states as well as switching speed can be strongly influenced by the application of an electrical current through the spin transfer torque effect. An external magnetic field can indeed modify the energy landscape. Through spin transfer torque effect, an electrical current also changes the stability of the states [START_REF] Slonczewski | Current-driven excitation of magnetic multilayers[END_REF] -as illustrated in Fig. 2(a) [START_REF] Rippard | Thermal relaxation rates of magnetic nanoparticles in the presence of magnetic fields and spin-transfer effects[END_REF], [START_REF] Li | Thermally assisted magnetization reversal in the presence of a spin-transfer torque[END_REF], [START_REF] Mizrahi | Magnetic stochastic oscillators: Noise-induced synchronization to under-threshold excitation and comprehensive compact model[END_REF]:

< τ AP/P >= τ 0 exp( ∆E kT (1 ± V V c )(1 ∓ H -H 0 H k ) n ), (3)
where V c is the junction critical voltage, V is the bias seen by the junction, H is the magnetic field along the junction easy axis, H k is its anisotropy and n is a real coefficient. H 0 is a residual field due to the fixed layer.

Although superparamagnetic junctions are extremely stochastic, they can exhibit very complex and interesting behaviors from a dynamic point of view. In particular, they exhibit a capability to synchronize to weak, subthreshold inputs [START_REF] Locatelli | Noise-Enhanced Synchronization of Stochastic Magnetic Oscillators[END_REF], [START_REF] Mizrahi | Controlling the phase locking of stochastic magnetic bits for ultra-low power computation[END_REF]. This form of synchronization, analogous to synchronization by stochastic resonance, exhibits surprising properties, as illustrated in Fig. 3. A subthreshold AC input, superimposed to white noise, is applied as a voltage to a superparamagnetic junction. Fig. 3 plots experimental measurements of the mean frequency of the junction as function of noise intensity and for different AC amplitudes. At low noise, the junction is not synchronized to the junction. At higher noise, synchronization is observed, that was therefore induced by noise. At even higher noise, this synchronization is lost. This behavior is perfectly explained by modeling, as we can see in Fig. 3 that a model based on equations (1-3) perfectly reproduces the experimental results. Superparamagnetic tunnel junctions, under appropriate condition, can also synchronize with each other, as studied in depth in [START_REF]Synchronization of electrically coupled stochastic magnetic oscillators induced by thermal and electrical noise[END_REF].

This behavior can offer highly promising prospects for unconventional circuits. As superparamagnetic tunnel junctions switch due to thermal noise, they have the potential to be used in circuits that consume very little energy (therefore harvesting some energy from thermal noise). Bioinspiration could provide a route as many schemes for computing (e.g. associative memories) rely on synchronization [START_REF] Izhikevich | Weakly pulse-coupled oscillators, fm interactions, synchronization, and oscillatory associative memory[END_REF], and multiple works have suggested that biology operates with stochastic resonance in some contexts [START_REF] Hänggi | Stochastic resonance in biology how noise can enhance detection of weak signals and help improve biological information processing[END_REF].

III. DETERMINISTIC SPINTRONIC NANOOSCILLATORS

Magnetic tunnel junctions can exhibit a very different type of oscillations. When appropriate conditions are met, the magnetization of a nanomagnet can precess, at frequencies ranging between 100M Hz to dozens of GHz. Many variations of such nanooscillators exist, here we focus on vortex based oscillators, introduced in Fig. 4 [4]. They consist in nanopillars composed of multiple magnetic and nonmagnetic layers, with a free magnetic layer in the middle and two pinned layers with orthogonal orientations. The bottom three layers implement a conventional magnetic tunnel junction. The remanent free layer magnetic state is called a vortex [START_REF] Guslienko | Magnetic vortex state stability, reversal and dynamics in restricted geometries[END_REF], where the magnetization curls in the free layer and goes perpendicular to the device in a center part (see Figure 4). The nanooscillator relies on a gyrotropic excitation, corresponding to a circular movement of the vortex [START_REF] Guslienko | Magnetic vortex state stability, reversal and dynamics in restricted geometries[END_REF] as seen in Fig. 4(b).

When an electrical current is applied to the junction, the different magnetic layers generate an effect on the vortex through spin-transfer [START_REF] Dussaux | Large amplitude spin torque vortex oscillations at zero external field using a perpendicular spin polarizer[END_REF]. If the current is above a certain threshold, the device demonstrates oscillations of its resistance: an oscillating voltage appears on the device. The frequency of the oscillators is determined by the applied current and is therefore tunable.

Different experimental physics studies have shown that this oscillation can be modeled by the Thiele equation [START_REF] Guslienko | Magnetic vortex state stability, reversal and dynamics in restricted geometries[END_REF], [START_REF] Thiele | Steady-state motion of magnetic domains[END_REF], where the whole device is modeled by the position of the vortex. Here, we present it with additional spin-transfer terms. X = ρe iχ is the position of the vortex (see Figure 4(b)). The modified Thiele equation reads:

iG Ẋ + D(ρ) Ẋ + (k(ρ) -iκ 1 I)X = κ 2 I + F noise (t) (4)
I is the electrical current, G is Gyrovector amplitude, D is the damping parameter, k is a confinement coefficient, ρ = |X|, κ 1 , κ 2 are spin transfer efficiencies [START_REF] Locatelli | Vortex-based spin transfer oscillator compact model for IC design[END_REF]. F noise (t) is a phenomenological stochastic force modeling fluctuations, based on fluctuation-dissipation theorem, as described in [START_REF] Tiberkevich | Temperature dependence of nonlinear autooscillator linewidths: Application to spin-torque nano-oscillators[END_REF]. The model should also include an amplitude limitation through a maximum speed for the vortex as detailed in [START_REF] Locatelli | Vortex-based spin transfer oscillator compact model for IC design[END_REF].

In parallel, the electrical resistance of the structure varies as:

R = R min 1 + 2 3 C TMR % Im(X) r (5)
R min is the minimum resistance of the device, TMR % its tunnel magnetoresistance, C = ±1, r is the radius of the device [START_REF] Locatelli | Vortex-based spin transfer oscillator compact model for IC design[END_REF]. Fig. 5 shows a simple simulation of this equation, in a situation where the input current I is constant and equal to 260µA. In Fig. 5(a), noise term F noise (t) is omitted, in Fig. 5(b) it is included. The frequency of the oscillation is 618MHz, which is consistent with the expected frequency for this particular device [START_REF] Guslienko | Magnetic vortex state stability, reversal and dynamics in restricted geometries[END_REF]. As seen in Figure 5(c), the noisy oscillator presents phase noise, corresponding to the linewidth of the peak that can be experimentally observed [START_REF] Dussaux | Large amplitude spin torque vortex oscillations at zero external field using a perpendicular spin polarizer[END_REF]. The linewidth extracted from simulations is 14.6MHz, while theory predicts 14.0MHz [START_REF] Tiberkevich | Temperature dependence of nonlinear autooscillator linewidths: Application to spin-torque nano-oscillators[END_REF]. It should be noted that this relatively high degree of noise can be strongly reduced in other geometries [START_REF] Tsunegi | High emission power and q factor in spin torque vortex oscillator consisting of feb free layer[END_REF].

Spin torque oscillators also possess a possibility to synchronize, to external sources, as well as to each other [START_REF] Locatelli | Efficient synchronization of dipolarly coupled vortex-based spin transfer nano-oscillators[END_REF]. This possibility is exhaled by the nonlinearity of the oscillators, and leads multiple groups to investigate bioinspired computing schemes that rely on the synchronization of spin torque oscillators [START_REF] Nikonov | Coupled-oscillator associative memory array operation for pattern recognition[END_REF]- [START_REF] Grollier | Spintronic nanodevices for bioinspired computing[END_REF].

IV. CONCLUSION

Although magnetic tunnel junctions are currently mostly known as a form of binary memristive device, we have seen that can also exhibit extremely rich oscillatory behaviors. These oscillators rely on a technology very similar to ST-MRAM, which has demonstrated CMOS compatibility and very large scale integration potential. Their use may therefore be envisioned in systems with millions of oscillators. Two separate regimes are possible, and we have proposed models in both regimes directly usable for circuit theory. The two regimes are extremely different, but both exhibit synchronization behaviors, which make them suitable for unconventional computation schemes, where the basic element (nanooscillator) provides considerably more functionality that traditional electron devices. This possibility interrogates a new way to design electronic circuits, and is especially relevant for bioinspired schemes, as biology itself relies on very rich nanodevices. Nevertheless, this idea comes with severe challenges, the most important being the need to deal with variability inherent to all nanodevices. Therefore, this work opens the way to important research to make electronics based on nanooscillators a reality.

Fig. 1 .

 1 Fig. 1. (a) Simplified illustration of a typical magnetic tunnel junction. (b) Illustration of the energy barrier of a magnetic tunnel junction, with two stable states and the influence of electrical current through spin transfer torque (STT).

Fig. 2 .

 2 Fig. 2. Superparamagnetic tunnel junctions acting as stochastic oscillator: (a) schematic and (b) experimental measurement of resistance as function of time. Experimental details are given in [7].

Fig. 3 .

 3 Fig.3. Synchronization experiment applied to a stochastic oscillator. A subthreshold AC signal with noise is applied on a stochastic oscillator. Mean frequency of the oscillator as function of noise level and for different AC amplitudes. Experimental details are given in[START_REF] Mizrahi | Controlling the phase locking of stochastic magnetic bits for ultra-low power computation[END_REF].

Fig. 4 .

 4 Fig. 4. (a) Schematic of a vortex oscillator. (b) Trajectory of the vortex during oscillation.

Fig. 5 .

 5 Fig. 5. Simulation of a vortex nanooscillator (a) without noise term and (b) with noise term. (c) spectrum of the simulation with noise.

ACKNOWLEDGMENT

This research was partly supported by Labex NanoSaclay (ANR-10-LABX-0035) and MEMOS (ANR-14-CE26-0021).


Oskars Ozolins
email: oskars.ozolins@ri.se

Francesco Da Ros

Valentina Cristofori

Xiaodan Pang

Aleksejs Udalcovs

Richard Schatz

Leif Katsuo Oxenløwe

Sergei Popov

Gunnar Jacobsen

Christophe Peucheret

Impact of Phase-Filtering on Optical Spectral Reshaping with Microring Resonators for Directly-Modulated 4-PAM Signals

Keywords:

come

Impact of Phase-Filtering on Optical Spectral Reshaping with Microring Resonators for Directly-Modulated 4-PAM Signals 1. Introduction

The need for high-speed connectivity in datacenter environments is driving a significant effort in researching compact, low-cost and energy efficient solutions [START_REF] Ozolins | 100 Gbaud 4PAM link for high speed optical interconnects[END_REF][START_REF] Shen | Integrated hybrid III-V/SOI directly modulated DFB laser and ring …[END_REF]. One of the proposed solutions relies on the use of directly modulated lasers (DMLs), thus replacing the need for external modulators. This choice, however, introduces the challenge of a limited transmission reach due to low dispersion tolerance and poor modulation extinction ratio (ER). These limitations may be overcome by optical spectral reshaping (OSR) by means of an optical filter, e.g an integrated microring resonator (MRR) [START_REF] Shen | Integrated hybrid III-V/SOI directly modulated DFB laser and ring …[END_REF]. Several demonstrations of OSR using MRRs have been reported for both on-off keying (OOK) [START_REF] Shen | Integrated hybrid III-V/SOI directly modulated DFB laser and ring …[END_REF][START_REF] Cristofori | Direct modulation of a hybrid III-V/Si DFB laser with MRR[END_REF] and 4-pulse amplitude modulation (PAM) [START_REF] Yang | IM/DD-based 112-Gb/s/lambda PAM-4 transmission using 18-Gbps DML[END_REF][START_REF] Da Ros | 4-PAM dispersion-uncompensated transmission with micro-ring …[END_REF] showing the improvement achieved in terms of reach extension. However, the impact of MRR-based filtering, in terms of the interplay between ER enhancement and added dispersion, is yet unclear. While the MRR amplitude transfer function provides frequencyto-amplitude modulation conversion, enhancing the modulation ER, its phase transfer function may provide additional narrow-band dispersion which could potentially impact the transmission reach.

In this paper we address this key aspect by a detailed numerical analysis of the OSR comparing amplitude-only and complex transfer functions (TFs) of a MRR. We show reach improvements over dispersion-uncompensated standard single-mode fiber (SSMF) links of approximately 95% thanks to optimized amplitude-only OSR for 20 GBaud 4-PAM signals. The reach is further increased up to 120% with use of the optimized complex TF.

System under investigation

Simulations (see setup Fig. 1 (a)) for transmission of 20 Gbaud directly-modulated 4-PAM modulation over SSMF were performed in VPItransmissionMaker TM [6]. Two data streams generated from 2 15 -1 pseudo random bit sequences with non-return-to-zero (NRZ) line coding were passively combined to create an electrical 4-PAM signal, which was applied to a directly modulated laser (DML). Single-mode laser rate equations using parameters from [START_REF] An | Advanced Functionalities for Highly Reliable Optical Networks[END_REF] were used. After the DML, the in-to-through TFs of an MRR were applied, either considering the amplitude-only or the full complex response. The MRR free spectral range (FSR) was set to 60 GHz and its power coupling coefficient (kp) was varied between 0.3 and 0.9. After photodetection, the signal was low-pass filtered with a 15 GHz 4 th order Bessel filter. Error counting (2 16 symbols) was used to calculate the bit-error-ratio (BER). shows the transmission reach without OSR of the 20 Gbaud 4-PAM signal with the DML biased at 150 mA and with a 50 mA modulation current swing. These biasing conditions are kept throughout the analysis. The quality factor (in linear scale, Qlinear) can be calculated from the BER. The transmission reach below the 7% overhead hard-decision forward error correction (HD-FEC) threshold level of Qlinear=2.67 (BER=3.8×10 -3 ) is approximately 5 km (0.5 km step length in simulations). The spectrum of the signal at the DML output is reported in Fig. 1(c) together with MRR transfer functions with kp= 0.3 and 0.5 in Fig. 1(d) and (e), respectively.

Results and discussions

Detailed simulations were performed to investigate whether the phase transfer function of the MRR used for OSR may provide additional reach increase for the directly-modulated 4-PAM signal through partial dispersion compensation. The OSR parameters, i.e. detuning of the MRR resonance from the signal central frequency and MRR power coupling coefficient (kp), were optimized at a fixed MRR FSR of 60 GHz and for 5 km transmission, corresponding to the DML transmission reach without OSR. The optimization was performed for both amplitudeonly and complex MRR TFs. From Fig. 2(a) it is found that the highest signal quality (Qlinear=3.6) is achieved with kp = 0.5 and a detuning of -6.7 GHz. With these optimized parameters, the transmission reach was investigated, as shown in Fig. 2(b). With amplitude-only OSR, the transmission reach increases to 9.5 km. The optimum frequency detuning was also checked, and it is the same for highest signal quality as in Fig. 2 (a). The improvement is attributed to the frequency-to-amplitude modulation conversion that enhances the modulation ER. The same procedure (optimization of OSR and transmission reach analysis) is then repeated considering the full complex TF of the MRR. From Fig. 2(c) the highest Qlinear=3.65 with kp of 0.3 and a detuning of -3.3 GHz is found. Fig. 2(d) shows a transmission reach increase to 11 km. This additional reach is attributed to narrow-band dispersion introduced by the complex TF of the MRR.

Conclusions

The transmission reach enhancement is numerically demonstrated for directly-modulated 4-PAM signals using a microring resonator-based optical spectral reshaping. The contribution of the MRR phase transfer function is investigated, showing a further reach increase compared to amplitude-only filtering.

5 MRR

 5 Fig. 1. (a) System under investigation; (b) Qlinear as a function of received power and transmission distance for 20 Gbaud 4-PAM with directly modulated laser; (c) spectrum for 20 Gbaud 4-PAM; MRR transfer function: (d) amplitude and (e) phase.

Fig. 1 (

 1 Fig.1(b) shows the transmission reach without OSR of the 20 Gbaud 4-PAM signal with the DML biased at 150 mA and with a 50 mA modulation current swing. These biasing conditions are kept throughout the analysis. The quality factor (in linear scale, Qlinear) can be calculated from the BER. The transmission reach below the 7% overhead hard-decision forward error correction (HD-FEC) threshold level of Qlinear=2.67 (BER=3.8×10 -3 ) is approximately 5 km (0.5 km step length in simulations). The spectrum of the signal at the DML output is reported in Fig.1(c) together with MRR transfer functions with kp= 0.3 and 0.5 in Fig.1(d) and (e), respectively.

7 Fig. 2 .

 72 Fig. 2. (a) Qlinear optimization for 20 Gbaud directly-modulated 4-PAM at 5 km distance with amplitude-only TF of MRR; (b) transmission reach increase with optimized amplitude-only TF of MRR; (c) Qlinear optimization at 5 km distance with complex TF of MRR; (d) transmission reach increase with optimized complex TF of MRR.

Acknowledgement

This work was supported by the Vetenskapsrädet grant no. 2016-04510 PHASE, the Swedish SRA ICT-The Next Generation project SCENE and EC Seventh Framework Program (grant no. 619626 SEQUOIA).


Mathilde Gay

Laurent Bramerie

Christophe Peucheret

John O'carroll

Richard Phelan

Michael Gleeson

Diarmuid Byrne

Brian A Kelly

Marta Nawrocka

Fabienne Saliou

Frequency Drift Reduction in a Four-Laser Array for TWDM PON Applications

Keywords: Laser array, TWDM passive optical network, frequency drift

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

IME and wavelength division multiplexing (TWDM) for next generation passive optical networks (NG-PON2) will require wavelength-tunable transmitters supporting burst mode operation at the customer side. Configurations with four pairs of wavelengths have been shown to meet the requirements of NG-PON2 [START_REF] Ma | Demonstration of a 40Gb/s time and wavelength division multiplexed passive optical network prototype system[END_REF]. In this configuration, the transmitter at each optical network unit (ONU) must be tunable to any of the four upstream wavelengths. In this context, laser arrays become attractive solutions combining fast wavelength tuning and cost constraints compatibility. The frequency drift of lasers under burst-mode operation is a concerning issue, as the laser emission frequency must remain within the multiplexer passband during laser switch-on regardless of the duration of the burst. In a 100-GHz grid scenario, a tolerance of +/-20 GHz for the laser emission frequency is specified in the standards [START_REF]ITU-T. 40-Gigabit-capable passive optical networks 2 (NG-PON2): physical media dependent (PMD) layer specification[END_REF]. Different solutions have been proposed so far for frequency drift reduction in single laser structures. These include integration of a heating element along the laser for counter-heating operation [START_REF] Fujiwara | Suppression of thermal wavelength drift in super-structure grating distributed Bragg reflector (SSG-DBR) laser with thermal drift compensator[END_REF][START_REF] Debrégeas | Quasi frequency drift suppression for burst mode operation in low-cost thermally-tuned TWDM-PON[END_REF], control of the burst ramp [START_REF] Simon | Experimental demonstration of low cost wavelength drift mitigation for TWDM systems[END_REF], sub-threshold heating [START_REF] Poehlmann | Wavelength drift of burst-mode DML for TWDM-PON[END_REF] or engineering of the signal density of the burst preamble [START_REF] Borkowski | TWDM PON preamble engineering for burst-mode frequency drift reduction through mitigation of a DFB turn-on blue shift[END_REF]. In this work, we present a new transmitter architecture based on an array of four single-mode lasers integrated with a multimode interference (MMI) coupler in a regrowth-free technology that is compatible with TWDM-PON cost requirements. This transmitter is to be placed at the ONU side to allow fast tunability, thus never emitting simultaneously on two wavelengths. Frequency drift under burst mode operation is addressed by taking advantage of the presence of the inactive lasers in the array that are biased below threshold in order to limit the active laser frequency drift when it is switched on to transmit a data burst. A maximum drift excursion below 20 GHz was measured for the lasers of the array thanks to the introduction of our proposed thermal management leading to 17% drift reduction. The drift under burst and wavelength switching operation is furthermore assessed for the first time to our knowledge with a measured drift reduction of 22%.

II. TRANSMITTER DESCRIPTION

Each laser of the array structure (represented in Fig. 1(a)) is based on the discrete-mode laser technology [START_REF] O'carroll | Wide temperature range 0 < T < 85 °C narrow linewidth discrete mode laser diodes for coherent communications applications[END_REF] in which ridge waveguide Fabry-Perot lasers are fabricated using a regrowthfree AlInGaAs/InP technology. Single-mode operation is achieved by introducing index perturbations distributed along the waveguide laser cavity leading to more than 40-dB sidemode suppression. The laser structures are combined to a common output waveguide thanks to a 4×1 multi-mode interference (MMI) coupler. At this stage of the fabrication process, the lasers are biased with direct current only and no data transmission is permitted. Since for the ONU transmitter in a TWDM PON context only one single wavelength is used at any given time, external modulation can be performed thanks to a single common external modulator. An electro-absorption modulator (represented within a dashed box in Fig. 1(a)) will be included in the next generation of devices. It is however important to notice that external data modulation performed in this way will not modify the frequency drift. This early stage prototype is hence fully suitable for the proposed burst operation study. The 41 MMI structure, made from the same layer stack as the lasers, needs to be biased for transparent operation. A microscope picture of the device is shown on Fig. 1(b).

The spectrum measured at the output of the MMI when all four lasers are biased with a current of 80 mA is represented in Fig. 2. Four signals with wavelengths of 1544.2 nm, 1549 nm, 1551.7 nm and 1556.2 nm are emitted with respective output powers of -1.25 dBm, 0.25 dBm, -17 dBm and -5 dBm. The respective lasing thresholds are 30 mA, 35 mA, 30 mA and 35 mA. The origin of the lower power of channel 3 at 1551.7 nm is some sub-optimum implementation of the MMI coupler. The channel spacing was not a key parameter in this early stage for frequency drift study and the chip was not with 100 GHz channel spacing. The laser array was placed in a butterfly package including a Peltier element to maintain the temperature at 25 °C. Finally, a four-laser driver board was developed, leading to a 10 to 90% switch-on rise time of 20 ns when the bias is changed from 0 to 70-mA.

III. BURST MODE OPERATION AND FREQUENCY DRIFT

MANAGEMENT

We study the frequency drift of the transmitter when each laser is individually switched on between 25 mA and 70 mA bias, for a standardized burst period of 125 µs and for a burst duration varying between 1% and 99% of the period. The frequency drift is measured using a method developed for time-resolved frequency chirp measurement based on the use of a Mach-Zehnder interferometer (MZI) as an optical frequency discriminator [START_REF] Laverdiere | A new method for measuring time-resolved frequency chirp of high bit rate sources[END_REF]. In this experiment (Fig. 3 (a)), the transfer function of an MZI with 200-GHz free spectral range is emulated using a programmable optical filter, thus providing enhanced stability compared to traditional fiber-based MZI implementations.

The frequency drift measurement method relies on the measurement of the intensity of the filtered signal for three different relative detunings of the emulated MZI transfer function with respect to the laser frequency. The laser frequency drift is converted to power fluctuations by the MZI transfer function, which will add to the laser emitted power variations in different ways, depending on the biasing of the MZI. Thanks to measurements of the power of the filtered signals at the three detunings corresponding to points A, B and C in Fig. 3(c), the time varying phase can be separated from the time-varying intensity. The frequency drift can then be calculated from the retrieved time-varying phase. In our implementation, the signal is detected by a 15-GHz photodiode (PD) and acquired in a real time oscilloscope (3 GHz bandwidth, 5 GS/s).

In a TWDM context, where thermal effects are under consideration, the laser frequency reference should be defined during the first emitted burst, which happens during the ONU initialization phase (ranging). During this phase, a short burst (preamble followed by a few bytes) is transmitted and the carrier frequency is tuned to the center of the passband of the multiplexer channel. In this study, the laser nominal center frequency is thus determined for a burst duration of 1% of the cycle (1.25 µs). Fig. 4(a) presents the measured frequency drift with respect to the nominal frequency of laser 1 when the burst duration varies. The longer the burst, the stronger the heating compared to the reference case of 1% burst duration and the larger the frequency drift, with a maximum drift of -22.6 GHz The inactive lasers of the transmitter were then used as heaters for laser 1. For this purpose, biasing signals complementary to that injected to laser 1 were applied to lasers 2, 3 and 4 with current values just below their threshold currents Ith (between 0 and 25 mA), so that the emitted power ratio between emitting laser (1 in this case) and heating lasers (2, 3 and 4 in this case) is maintained above 30 dB (Fig. 3(b)). Fig. 4(b) depicts the resulting frequency drifts, which remain below 14 GHz. The compensation limits the temperature gradient regardless of the burst duration.

Identical measurements were performed on lasers 2 and 4. The frequency drift of laser 3 was not measured due to its lower output power. The values of maximum drift excursions (drift variation during the burst), with or without drift management, measured as a function of burst duration are summarized in Fig. 5. Laser 4 presents the largest drift, which reaches -24.2 GHz for long bursts. Using the proposed management, the drift is reduced below -20 GHz, making the drift excursion compliant with standard recommendations.

IV. SWITCHING BETWEEN TWO WAVELENGTHS

The frequency drift behavior when the transmitter is switched between laser 1 to laser 4 is now investigated. The command signals switch the bias currents of laser 1 and laser 4 between 25 mA and 70 mA with a burst duration of 62.5 µs and a burst cycle of 125 µs, as illustrated in Fig. 6(a). The frequency drift under this switching operation is represented in Fig. 6(b). The frequency drift of laser 1 is then measured as a function of burst duration when laser 4 burst duration is fixed. Three different cases of laser 4 burst of 1%, 50% and 99% of the cycle duration are considered (Fig. 6(c)). A maximum drift excursion of -27.3 GHz was measured when the burst length is 99% of the cycle duration for both lasers.

Drift reduction by adjusting bias currents of lasers 2 and 3 is now applied. The chronogram of lasers 2 and 3 bias currents is obtained by operating a NOR gate on laser 1 and 4 currents (Fig. 6(a)) with a current excursion between 0 and 25 mA. Using this drift management technique, a 22% drift excursion reduction is obtained, leading to a maximum of -21.2 GHz when the bursts applied to lasers 1 and 4 last 99% of the cycle duration. The study has been performed on laser presenting the largest frequency drift and which is located on the side of the array. A specific design minimizing the distance between lasers could probably enhance the heating effect of inactive lasers and possibly limit even more the frequency drift. Moreover, the possibility to integrate a semiconductor optical amplifier at the output of the laser was demonstrated in [START_REF] Carroll | Photonic Integrated Circuits for NGPON2 Tunable ONUs[END_REF]. This would increase the lasers output powers thus allowing to work at lower bias currents and consequently lead to smaller frequency drifts. Let us finally notice that the present study has been performed at 25°C; however, this kind of laser has been shown to operate in a large temperature range from 0 to 85°C. At temperatures where the lasing threshold evolves, the bias current of active and inactive lasers should be adjusted, both to maintain a constant output power of the active laser and to place the inactive lasers just below their thresholds.

V. CONCLUSION

The frequency drift of a four-laser array under burst and wavelength switching operation is assessed for TWDM-PON applications. The four-single-mode laser array integrated with an MMI coupler is based on a regrowth-free technology that is compatible with TWDM-PON cost requirements. The use of a laser array offers a double advantage: firstly it limits the required thermal variation for wavelength adjustment (divided by 4 in the case of a 4-laser array); secondly, we take advantage of the presence of inactive lasers in the array to limit the frequency drift of the active laser during burst emission. It results in a maximum frequency drift excursion below 20 GHz, which is compliant with TWDM-PON standards. The implemented drift management under burst and wavelength switching operation allows a reduction of up to 22% of the frequency drift for a laser located at the side of the array.

  Manuscript received March 3rd, 2018. This work was supported in part by the European Union (EUROSTARS projects E!10309 and E!8598).M. Gay, L. Bramerie, C. Peucheret are with Univ Rennes, CNRS, FOTON -UMR 6082, F-22305 Lannion, France (e-mail: mathilde.gay@enssat.fr).

Fig. 1 .

 1 Fig. 1. (a) Schematic representation of the 4-laser array structure. Only the part of the device within the solid box was implemented in this work. (b) Microscope picture of the device.

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. Spectrum at the output of the laser array when all lasers are biased at 80 mA. 1545 1550 1555

Fig. 6 .

 6 Fig. 6. (a) Laser driving currents chronograms in the case of wavelength switching. (b) Frequency drift excursions of lasers 1 and 4 with respect to their nominal center frequencies. (c) Frequency drift excursion, with and without compensation, of laser 1 as a function of the burst duration applied to this laser for different burst durations applied to laser 4.

  Fig. 5. Drift excursion versus burst duration of lasers 1, 2 and 4 with and without the proposed drift management.

	Drift excursion (GHz)	0 10 20 30				Laser Laser Laser Laser Laser Laser	1 1, 2 2 with with drift drift management management 4 4, with drit management
		0	50		100
		Burst	duration	(%	of	cycle)
						(GHz)	Laser 4 burst duration: 1% Laser 4 burst dur.: 1%, with drift management Laser 4 burst dur.: 50%
						Laser 4 burst dur.: 50%, with drift manag.
						Laser 4 burst dur.:99%
						Laser 4 burst dur.: 99%, with drift manag.

J. O'Carroll, R. Phelan, M. Gleeson, D. Byrne, B. Kelly and M. Nawrocka are with Eblana Photonics, 3 West Pier Business Campus, Dun Laoghaire, Co. Dublin, A96 A621, Ireland (john.ocarroll@eblanaphotonics.com).F. Saliou is with Orange Laboratory, 22300 Lannion, France (fabienne.saliou@orange.com).Frequency Drift Reduction in a Four-Laser Array for TWDM PON ApplicationsMathilde Gay, Laurent Bramerie, Christophe Peucheret, John O'Carroll, Richard Phelan, Michael Gleeson, Diarmuid Byrne, Brian Kelly, Marta Nawrocka, and Fabienne Saliou


Karl Joulain

Maximum thermal emission of subwavelength spherical objects

published or not. The documents may come

Maximum thermal emission of subwavelength spherical objects

FIG. 1: Nanoparticle at temperature T with a radius much smaller than the thermal wavelength λ T .

I. SYSTEM CONSIDERED

We consider a spherical object filled with a material which permittivity is given by and temperature is given by T . The system is depicted in Fig. 1. The simple question which is adressed in this note is to know what is the maximum thermal emission of such a spherical object when its size is much smaller than the thermal wavelength λ T given by the Wien law and corresponding to the wavelength at which Planck emission is maximum (λ T T = 2898 µm K).

II. MAXIMUM THERMAL EMISSION OF A DIPOLAR NANOPARTICLE

Let us now calculate the thermal emission of a spherical object. A consequence of the reciprocity theorem in electromagnetism is the Kirchhoff law stating that the absorptivity of any object is equal to its emissivity. Thermal emission φ ω in the most genreal case is thus the product of the absorption cross section σ abs and of the so called thermal emittance

H 0 ω = ω/[4π 2 c 2 (exp[ ω/k b T ])] φ ω = σ abs H 0 ω = ω 3 4π 2 c 2 (exp[ ω/k b T ] -1) σ abs (1)
where, ω is the angular frequency considered, is the reduced Planck constant, c the velocity of light and k b the Boltzmann constant. In the case of an isolated sphere, the absorption cross section can be obtained exactly with Mie theory. When this sphere has a radius which is much smaller than the wavelength considered, one is in the dipolar approximation and the absorption cross section is completely dominated by the first term given by the Mie theory.

Grigoriev et al.

[1] have shown that it is possible to achieve an ideal absorption if the sphere permittivity follows a very specific function on ρ = ωR/c where R is the sphere radius.

When this condition is fulfilled, the absorption is maximized and the maximum absorption cross section is given by

σ max abs = 5π k 2 0 ( 2
)
where k 0 = ω/c. Note that under these conditions, the scattering cross section is equal to the absorption cross section. Reporting the maximum scattering cross section in the expression of the flux and integrating over all the frequencies, the maximum flux emitted by a spherical object in the dipolar approximation is

φ max = 5πk 2 b T 2 24 = 1.18 × 10 -12 W K -2 T 2 (3)
which is apart to a numerical number the product of the quantum of thermal conductance with the temperature. Note that this flux does not depend on the sphere size (although the optimized dielectric function maximizing absorption cross-section σ abs depends on it) and that it scales as T 2 .

[1] V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, ACS Photonics 2, 263 (2015).