Mehdi Alouini
email: mehdi.alouini@univ-rennes1.fr

Gwennael Danion

Marc Vallet

Ultra-narrow linewidth self-adaptive photonic oscillator: principle and realization

Highly coherent optical sources are a key element in several fields of physics, in particular in time frequency metrology. Over the past decennia, there has been particular efforts in developing such sources to the expense of sophisticated laser systems and relatively smart electronics. We propose here a new general principle of a self-adaptive oscillator where the intricate operation of a 100-m-long active optical resonator and a standard semiconductor laser offers a very high spectral purity and can be tailored to any wavelength. Single frequency operation of this self-adaptive photonic oscillator is achieved without any servo locking or stabilization electronics. Free running operation leads to a Lorentzian linewidth of 40 mHz. The long-term drift of the optical frequency in the free running regime is within 10 MHz over hours. This principle applies to any wavelength attainable by laser diodes which opens tremendous opportunities in particular in applications where atomic or molecular transitions require precise wavelengths.

Introduction

Owing to their major role in most modern physics experiments, single-frequency narrow-linewidth lasers are the subject of intensive research efforts. They are becoming mandatory for a wide field of applications including microwave photonics [START_REF] Capmany | Microwave Photonics Combines Two Worlds[END_REF][START_REF] Tetsumoto | Optically Referenced 300 GHz Millimetre-Wave Oscillator[END_REF][START_REF] Zheng | Optical microwave generation with an ultrastable Fabry-Perot cavity in a laser diode self-injection loop[END_REF] , time-frequency metrology [START_REF] Brodnik | Optically synchronized fibre links using spectrally pure chip-scale lasers[END_REF] , atom manipulation [START_REF] Ahmadi | Observation of the 1S-2S transition in trapped antihydrogen[END_REF] , atomic clocks [START_REF] Jiang | Making optical atomic clocks more stable with 10 -16 -level laser stabilization[END_REF][START_REF] Vogt | Demonstration of a transportable 1 Hz-linewidth laser[END_REF][START_REF] Hinkley | An atomic clock with 10 -18 instability[END_REF][START_REF] Loh | Operation of an optical atomic clock with a Brillouin laser subsystem[END_REF] , high resolution spectral analysis, resonant gyroscopes, coherent communications [START_REF] Kikuchi | Coherent Optical Communications: Historical Perspectives and Future Directions[END_REF][START_REF] Yoshida | 64 and 128 coherent QAM optical transmission over 150 km using frequency-stabilized laser and heterodyne PLL detection[END_REF][START_REF] Yoshida | 10 channel WDM 80 Gbit/s/ch, 256 QAM bi-directional coherent transmission for a high capacity next-generation mobile fronthaul[END_REF] , Lidar [START_REF] Karlsson | Linearization of the frequency sweep of a frequency-modulated continuous-wave semiconductor laser radar and the resulting ranging performance[END_REF] , coherent long-haul sensing, gravitational wave detection [START_REF] Bondu | Ultrahigh-spectral-purity laser for the VIRGO experiment[END_REF][START_REF] Kwee | Stabilized high-power laser system for the gravitational wave detector advanced LIGO[END_REF][START_REF] Bode | Advanced LIGO Laser Systems for O3 and Future Observation Runs[END_REF] , quantum optics, and more generally in research involving light-matter interactions [START_REF] Daffurn | A simple, powerful diode laser system for atomic physics[END_REF] . Over the past decades, several approaches have been pursued in view of generating the most coherent possible electromagnetic emission. In stringent applications such as atom manipulation or time-frequency metrology, external cavity semiconductor lasers are currently widely used due their scalability in terms of wavelength, the external cavity configuration improving their frequency noise performances [START_REF] Ye | Tunable External Cavity Diode Lasers[END_REF][START_REF] Daffurn | A simple, powerful diode laser system for atomic physics[END_REF] . Such lasers are then often stabilized on Fabry-Perot cavities [START_REF] Young | Visible Lasers with Subhertz Linewidths[END_REF][START_REF] Kessler | A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity[END_REF][START_REF] Matei | 1.5 μm Lasers with Sub-10 mHz Linewidth[END_REF] , atomic transitions [START_REF] Zhang | Ultranarrow linewidth photonic-atomic laser[END_REF] or optical frequency combs [START_REF] Jin | Towards generation of millihertz-linewidth laser light with 10 -18 frequency instability via four-wave mixing[END_REF][START_REF] Reynolds | Optical frequency stabilization with a synchronous frequency-tovoltage converter[END_REF] . As compared to monolithic semiconductor lasers, solid-state [START_REF] Kane | unidirectional single-mode Nd:YAG ring laser[END_REF][START_REF] Byer | Diode Laser-Pumped Solid-State Lasers[END_REF][START_REF] Danion | Dual frequency laser with two continuously and widely tunable frequencies for optical referencing of GHz to THz beatnotes[END_REF] and fiber [START_REF] Fu | Review of recent progress on single-frequency fiber lasers[END_REF][START_REF] Zhang | Ultra-narrow linewidth full C-band tunable single-frequency linear-polarization fiber laser[END_REF] . lasers offer intrinsic narrow linewidths. Moreover, these lasers are free from amplitude to phase noise conversion, which make them appreciated in demanding applications where stable and high spectral purity is necessary such as for gravitational wave detection [START_REF] Bondu | Ultrahigh-spectral-purity laser for the VIRGO experiment[END_REF][START_REF] Kwee | Stabilized high-power laser system for the gravitational wave detector advanced LIGO[END_REF] . Among solid state lasers, Brillouin fiber lasers have attracted a peculiar attention [START_REF] Chen | Low-noise, single-frequency, single-polarization Brillouin/erbium fiber laser[END_REF][START_REF] Chen | 53-dB phase noise suppression and Hz-range linewidth emission in compact Brillouin/erbium fiber laser[END_REF][START_REF] Loh | Ultra-narrow linewidth Brillouin laser with nanokelvin temperature self-referencing[END_REF][START_REF] Gundavarapu | Sub-Hz linewidth photonic-integrated Brillouin laser[END_REF][START_REF] Debut | Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers[END_REF] . Indeed, the Stimulated Brillouin Scattering (SBS) offers extremely narrow gain bandwidth which facilitates single mode oscillation of a long optical resonator favoring a natural narrow linewidth [START_REF] Grudinin | Brillouin Lasing with a CaF 2 Whispering Gallery Mode Resonator[END_REF] .

In addition to those inherent narrow linewidth lasers, active or passive stabilization schemes are usually employed to further stabilize the laser line and/or to narrow down its linewidth. In this regard, self-referenced frequency combs became the standard and undoubtedly the most powerful stabilization tool that is employed as a ruler which enables to stabilize the single mode laser frequency and to narrow down its linewidth at the same time. Obviously, sophisticated phase locked loops whose bandwidth is higher than the laser linewidth must be employed. Alternatively, locking the single mode laser to an UltraLow Expansion glass cavity (ULE) that is acoustically insulated, thermalized, and put into vacuum has been shown to be a very attractive approach as well [START_REF] Kessler | A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity[END_REF][START_REF] Matei | 1.5 μm Lasers with Sub-10 mHz Linewidth[END_REF] . Indeed, as compared to atomic or molecular lines the ULE cavity offers thousands of possible stable optical lines. A Pound-Drever-Hall servo loop is then commonly used to lock the laser frequency to one of the ULE cavity resonances. In all cases single mode laser has to exhibit a narrow linewidth in order to release the stringent constraints on the servo-locking electronics. Accordingly, the development of compact and ultra-narrow linewidth lasers is still today a very hot research topic [START_REF] Liang | Ultralow noise miniature external cavity semiconductor laser[END_REF] . With this aim in view, several linewidth narrowing techniques have been demonstrated more recently. They generally rely on optical reinjection of the single mode laser using external Bragg grating reflectors, micro-resonators [START_REF] Galiev | Spectrum collapse, narrow linewidth, and Bogatov effect in diode lasers locked to high-Q optical microresonators[END_REF] or even Rayleigh scattering in long fibers [START_REF] Huang | Linewidth suppression mechanism of self-injection locked single-frequency fiber laser[END_REF][START_REF] Kasai | 1.5 μm, mode-hop-free full C-band wavelength tunable laser diode with a linewidth of 8 kHz and a RIN of -130 dB/Hz and its extension to the L-band[END_REF][START_REF] Ishii | Spectral linewidth reduction in widely wavelength tunable DFB laser array[END_REF][START_REF] Huang | Tens of hertz narrow-linewidth laser based on stimulated Brillouin and Rayleigh scattering[END_REF] .

In parallel, in the microwave domain, an important step has been taken in achieving high spectral purity microwave oscillators [START_REF] Maleki | The optoelectronic oscillator[END_REF] . Accordingly, Opto-Electronic Oscillators (OEO) are to date the simplest, most efficient and reliable microwave resonators. They rely on the use of long-fiber optical delay lines in order to bring the microwave resonator length to kilometer scale [START_REF] Yao | Optoelectronic oscillator for photonic systems[END_REF] . Single mode operation is ensured by a microwave filter, whereas the linewidth is ruled by the long fibered delay line [START_REF] Vallet | Self-Stabilized Optoelectronic Oscillator Using Frequency-Shifted Feedback and a Delay Line[END_REF][START_REF] Zeng | Stable and Finely Tunable Optoelectronic Oscillator Based on Stimulated Brillouin Scattering and an Electro-Optic Frequency Shift[END_REF] . Interestingly, OEOs do not require any servo locking electronics as the optical delay line is part of the oscillator itself.

We report here on a Self-Adaptive Photonic Oscillator (SAPO) [START_REF] Alouini | Spectral narrowing module, refined spectral line device and method therefor[END_REF] whose operation principle in the optical domain is inspired by the OEO paradigm in the microwave domain. Accordingly, the extremely high spectral purity obtained in the free running regime, without any servo-locking, contrasts with any other reported approach for generating narrow linewidth optical waves.

SAPO principle

Extending the OEO principle in the optical domain relies on the possible design of a long resonant loop which operates itself in the optical domain. Fig. 1 illustrates the principle of an OEO in comparison to what should be a SAPO. The main ingredients for realizing an OEO are (i) a filtering stage which selects the central microwave frequency (ii) a long optical delay line which provides the high spectral purity when embedded within the microwave resonator, and (iii) the amplifying stage which compensates for the losses mainly generated by microwave-to-optical and optical-to-microwave conversions. It is worthwhile to notice that a narrow bandwidth microwave filter is mandatory in order to ensure single frequency oscillation. By moving to the optical domain, i.e. at around 200 THz frequencies, this constraint becomes a challenging aspect. SBS effect is the cornerstone for building a SAPO. Indeed, this physical effect, when induced in a single mode silica fiber, enables optical amplification over an extremely narrow bandwidth, i.e. of around 30 MHz [START_REF] Norcia | Efficient single-mode Brillouin fiber laser for low-noise optical carrier reduction of microwave signals[END_REF][START_REF] Vidal | Tunable and reconfigurable photonic microwave filter based on stimulated Brillouin scattering[END_REF] . Accordingly, the filtering function as well as the amplification function can be achieved together. Nevertheless, SBS requires, firstly, a sufficiently long time/distance light-matter interaction and needs, secondly, to be activated with a high intensity and narrow linewidth pump beam. To this aim, Brillouin oscillation is usually realized in micro-resonators in order to take advantage of their high-quality factor or in long fiber resonators provided that their free spectral range is lower than the inherent 30 MHz SBS gain bandwidth. In all cases the high spectral quality of the pump beam is a key condition. More importantly, an electronic servo-locking apparatus is required to efficiently feed the Brillouin resonator [START_REF] Loh | Ultra-narrow linewidth Brillouin laser with nanokelvin temperature self-referencing[END_REF][START_REF] Danion | Brillouin assisted optoelectronic selfnarrowing of laser linewidth[END_REF] .

Fig. 1:

Parallel between the operation principle of an OptoElectronic Oscillator (OEO) and a Self-Adaptive Photonic Oscillator (SAPO). The SAPO can be considered as an OEO extension in the optical domain.

For realizing an SAPO, these two constraints must be released. Let us consider that the pump laser and the Brillouin resonator constitute respectively the primary and secondary optical resonators, where the primary resonator provides the required energy, and the secondary long resonator generates a single mode high spectral purity electromagnetic field. To make the SAPO operation possible the two resonators must be optically intertwined so that they behave from an electromagnetic point of view as a single resonator. This single-resonator condition implies, firstly, that the secondary long resonator must be open (i.e. non-resonant) regarding the field coming from the primary resonator. Conversely, it has to be closed (i.e. resonant) for the generated electromagnetic field so as to provide the targeted high spectral purity. This can be fulfilled using a non-reciprocal resonator which is, by construction, non-resonant clockwise regarding the incoming field but resonant counterclockwise. The single-resonator condition implies also that the electromagnetic field generated by the second resonator must be able to easily couples into the primary resonator. Thus, counterintuitively, the primary resonator has to exhibit a very low-quality factor. This property is commonly satisfied in semiconductor lasers. Consequently, the overall constraints to realize the proposed SAPO have practical solutions, making it conceivable form a technical point of view.

SAPO practical realization

In practice the above-mentioned ingredients to realize a self-adaptive photonic-oscillator can be gathered together starting from a standard single mode DFB semiconductor laser which pumps a long Brillouin resonator designed to be non-resonant for the pump field but resonant for the Stokes wave as depicted in Fig. 2(a). The generated contra-propagating Stokes-wave, which in turn experiences the resonant condition, presents a high spectral purity. This Stokes wave is then reinjected to the DFB laser after being frequency up-shifted to match the pump central frequency (Fig. 2(b)). Provided that this frequency shift is precisely equal to the Brillouin offset, the DFB laser frequency will acquire the spectral properties of the reinjected field and the resonant condition of the whole photonic oscillator is accordingly satisfied. Hence, after one round-trip the linewidth of the DFB laser narrows down making the Brillouin process even more efficient leading to a narrower Stokes-wave linewidth which is going to reinject the DFB laser once again, and so on. 2), the up-converted Stokes wave is not reinjected into the DFB pump laser. (2) 100 times average of (1). From (3) to (6), time evolution of the beatnote when the up-converted Stokes wave is reinjected into the DFB pump (with 100 times average). The beatnote shrinks while its power spectal density increases showing that the energy originally contained in (2) concentrates in this narrow line.

This cascade effect, schematized in Fig. 2(c), takes place inherently. It implies a dramatic linewidthreduction of the DFB laser while the Brillouin process efficiency is enhanced. After a few round-trips, the SAPO reaches its stationary regime where the DFB and Stokes-wave linewidths are equal.

In practice, the 100-m-long Brillouin loop is closed using an optical circulator that ensures a resonant condition for the counter-clock-wise Stokes-wave while offering a single pass condition for the clockwise pump wave. This enables the whole pump energy to enter into play whatever the initial pump linewidth. In our proof of concept, the DFB laser oscillates at 1550 nm and delivers 20 mW output power. Half of its power is sent into a home-made low power erbium doped fiber amplifier which raises the output power to 100 mW before entering the fiber loop. The remaining power 10 mW serves as useful signal. The free-running linewidth of this laser has been measured to be 4 MHz, i.e., four times larger than the Free Spectral Range (FSR) of the nonreciprocal Brillouin resonator. Owing to the homogeneous nature of SBS gain, single frequency and mode hop free operation of the Stokes wave can be obtained provided that one resonance of the Brillouin resonator matches the maximum of the SBS gain [START_REF] Danion | Mode-hopping suppression in long Brillouin fiber laser with non-resonant pumping[END_REF] . This condition is fulfilled by up-shifting the Stokes-wave frequency by the amount corresponding to the Brillouin frequency offset, i.e., 10.9683 GHz in our case. To this aim, part of the Stokes wave is extracted and phase modulated at 10. 9683 GHz before being reinjected to the DFB laser (Fig. 2(b)). Among the two generated sidebands, the upper sideband frequency will coincide with the pump frequency. The two remaining tones being far from the DFB laser oscillating frequency, they are filtered out. Thus, the DFB laser locks to the up-shifted Stokes-wave and acquires its spectral purity (Fig. 2(d)). More importantly, this DFB-laser being itself the pump source generating the Stokes-wave, the locking mechanism is self-adaptive as it will be discussed later. The oscillation process evolves after a few round-trips towards a stationary state where the pump carries the same spectral purity as the Stokes wave without any servo-locking. When turned on, the SAPO do not exhibit any mode hopping or frequency instabilities or intensity fluctuations as long as it is turned on, i.e., during days.

Free running spectral characteristics

Spectral purity

The SAPO has been assembled in a compact and thermalized housing (Fig. 3(a)). The frequency noise is measured using a delayed self-heterodyne apparatus based on an unbalanced Mach-Zehnder interferometer including a 80 MHz acousto-optic modulator on one and a 700 m long fiber on the other arm. The self-heterodyne beatnote spectrum is then measured around 80 MHz with an electrical spectrum analyzer. An overview of the self-adaptive linewidth narrowing is given in Fig. 2(b) where the self-heterodyne beatnotes, for the DFB outside and inside the SSPO, are displayed with the same scales for a span of 150 MHz. The latest beatnote is zoomed in in Fig. 2(c) up to a span width of 50 kHz.

In order to extract comprehensive results this beatnote is now measured with a Rohde & Schwarz RS-FSW26 phase noise analyzer. The acquired phase noise is then converted into frequency noise taking into account the interferometer response [START_REF] Mercer | 1/f Frequency Noise Effects on Self-Heterodyne Linewidth Measurements[END_REF][START_REF] Camatel | Narrow Linewidth CW Laser Phase Noise Characterization Methods for Coherent Transmission System Applications[END_REF] . Moreover, to cover the 0.01-100 kHz bandwidth, and due to the low levels of noise obtained here, it was necessary to acquire the phase noise using two fiber lengths in the unbalanced interferometer, that is 700 m and 5 km for respectively the high and low frequency parts of the spectrum. As shown in the composite spectrum of Fig. 3(d), the SAPO frequency noise is 100 Hz 2 /Hz, 2 Hz 2 /Hz and 0.03 Hz 2 /Hz at 0.1 kHz, 1 kHz and 10 kHz frequency offset respectively. This spectrum indicates that the SAPO intrinsic linewidth is at around 40 mHz. Moreover, the -separation line, conventionally used to assess the flicker noise [START_REF] Di Domenico | Simple Approach to the Relation between Laser Frequency Noise and Laser Line Shape[END_REF] , reveals a 200 Hz linewidth only for 0.1 s integration time. This result is the state of the art for a free-running and servo-free oscillator. It confirms the high potential of the proposed SAPO principal for realizing ultranarrow linewidth optical sources.

Intensity noise characteristics

The SAPO includes by principle an extremely long delay line imbedded into the resonant system. Taking into account the propagation length in the SAPO, including the Brillouin loop, the effective photon lifetime is predicted to be in the µs range. This photon lifetime being significantly larger than any other time constant of the system, class A operation is expected [START_REF] Baili | Direct observation of the class-B to class-A transition in the dynamical behavior of a semiconductor laser[END_REF] , meaning that the SAPO intensity noise should not exhibit any resonant excess noise and should be shot noise limited. The relative intensity noise (RIN) has been measured at high (Fig. 4(a)), medium (Fig. 4(b)) and low (Fig. 4(c)) frequencies using dedicated RIN benches [START_REF] Kerchaoui | Electrically pumped shot-noise limited class A VECSEL at telecom wavelength[END_REF] . As shown in the different RIN spectra of Fig. 4, the SAPO exhibits a white noise whose level is -160 dB/Hz from 100 kHz to 18 GHz. This level corresponds to the shot noise floor for 5 mW detected optical power. It is worthwhile mentioning that we have deliberately limited the detected power to 5 mW in order to ensure that the RIN measurement chain operate within its linear region. Even though shot noise limited operation was not the main goal for developing this SAPO, it actually overcomes any other class A laser reported in the literature [START_REF] Ghaya | Evidence of Ultra Low Microwave Additive Phase Noise for an Optical RF Link Based on a Class-A Semiconductor Laser[END_REF] . Indeed, an equivalent optically pumped class A laser would show an excess noise plateau due to the pump noise up to 100 kHz [START_REF] Pes | Class-A Operation of an Optically-Pumped 16 Μm-Emitting Quantum Dash-Based Vertical-External-Cavity Surface-Emitting Laser on InP[END_REF] .

Frequency stability

In order to characterize the SAPO long-term frequency drift in the free running regime, a second identical SAPO has been built. Their Brillouin-fiber-loop is thermalized using a thermoelectric controller. Moreover, they are properly insulated from acoustic vibrations. The two SAPOs have been positioned in two different experiment rooms so that they experience uncorrelated environmental perturbations. By adjusting the temperature of fiber spool resonator, the wavelength of the second SAPO is set at 0.16 nm, i.e. 20.2 GHz, from the first SAPO. Their beatnote is then acquired using a fast photodiode connected to the electrical spectrum analyzer. A beatnote spectrum between the two SAPOs with a span width of 300 kHz is illustrated in Fig. 5(a). Owing to the fact that the two SAPOs are free running, a random walk ranging from 10 to 20 MHz is observed for an acquisition time of 4 hours. The standard Allan deviation is also reported in Fig. 5(b). It can be noticed that it reaches a plateau above an integration time of 100 s. To go further in terms of characterization, the coming step will be the stabilization of such SAPO on molecular transitions.

Discussion on SAPO distinctive characteristics Why Stimulated Brillouin Scattering?

The first idea that comes in mind to obtain a narrow linewidth photonic oscillator is to realize the longest possible laser cavity. Indeed, the Schawlow Townes linewidth, i.e. the quantum limited linewidth, is inversely proportional to the photon lifetime inside the cavity. The longer the cavity, the narrower the linewidth is. Unfortunately, when the laser cavity exceeds a few meters, single mode operation becomes difficult to ensure due to the lack of sufficiently narrow and stable filters that can be inserted into the laser cavity. Furthermore, assuming that such a filter exists, an active tracking would be necessary to avoid mode hopping. This major constraint explains the fact that current techniques for narrowing down laser linewidths involve external delayed filtering components that are not part of the optical oscillator itself. With this respect, embedding the SBS effect within the optical oscillator brings the required narrow linewidth optical filtering function and also the self-adaptive function releasing the constraint of servo-locking. Finally, owing to the homogeneous spectral broadening of the SBS gain [START_REF] Danion | Mode-hopping suppression in long Brillouin fiber laser with non-resonant pumping[END_REF] , the length of the Brillouin resonator can exceed 10 times that which provides one mode within the Brillouin gain. As a result, a Schawlow Townes linewidth below 1 µHz is reachable.

Why a large linewidth a pump laser?

In the proposed SAPO architecture the pump laser must be efficiently sensitive to the Stokes wave, that is fed back, in order to act as part of the overall oscillator. Consequently, its intrinsic linewidth must be wide enough to make the SAPO oscillation robust against environmental perturbations. This condition is fulfilled by any standard semiconductor lasers. In return the Brillouin resonator has to be non-resonant for the pump so that any change in the spectral characteristics of the pump do not affect the pumping efficiency, bringing the SAPO to a self-adaptive oscillation.

Why SAPO operation contrasts with standard frequency-locked lasers?

The frequency robustness of the SAPO relies on the interleaved interaction between the DFB laser and the active Brillouin resonator. Its behavior cannot be understood considering these two elements separately. Indeed, let us assume that the frequency of the Stokes wave drifts due to a change of the fiber optical index. The DFB pump laser being injected by the Stokes wave, its frequency, and consequently the maximum of the induced SBS gain, will automatically follow this drift (Fig. 6(a)). As a result, no mode hops occur. Such a frequency dragging effect can be seen in Fig. 6(b) where the temperature of the Brillouin fiber was linearly increased. The SAPO frequency changes continuously over a span of 1 GHz, that is 500 times the fiber resonator FSR without any mode hopping. By contrast, if a perturbation is now applied to the DFB current, its frequency remains unchanged since this frequency is locked by the Stokes wave frequency, i.e., the fiber resonator. This second mechanism is shown in Fig. 6(b) where the SAPO frequency stays constant while the DFB laser current is varied over 1.5 mA corresponding to 1.5 GHz of frequency variation of the out of loop DFB. Such perturbation is equivalent to 700 FSRs of the fiber resonator. These results show clearly that the overall stability of the SAPO is ruled by the stability of the long fiber Brillouin resonator. Moreover, they highlight that the observed 10-20 MHz frequency random walk is consistent with the 10 mK thermalization precision of the fiber spool.

Conclusion

A self-adaptive photonic oscillator based on the OEO paradigm is proposed. Its operation principle is presented and the key elements for its realization are emphasized. This system exhibits an extremely low frequency noise without any electronic servo loop. The Lorentzian linewidth is measured to be 40 mHz and the flicker noise linewidth is 200 Hz for 0.1 s integration time in free running condition. The long-term drift of the optical line is within 10 MHz over a few hours. Moreover, its intensity noise turned to be limited by the shot noise. Plug and play prototypes of this SAPO have been already developed in the laboratory. Their performances remain unchanged for operating temperatures ranging from -10°C to 40°C. Fine tunability can be achieved by stretching or thermo-controlling the Brillouin fiber resonator, whereas coarse tunability can be obtained by electrically or thermally controlling the DFB laser. Current developments address its locking to molecular references. Owing to its high spectral performances in a self-sustained manner to its operation robustness and to its implementation simplicity, SAPOs open a wide avenue to all the fields where narrow linewidth lasers are requested. All the more so SAPO principle can be easily extended to any wavelength between 700 nm and 2 µm where fibered components and semiconductor are currently available.

Fig. 2 :

 2 Fig. 2: SAPO operation principle. (a) SAPO architecture. The modulator is modulated at the mean Brillouin phonon frequency f LO . (b) The non-resonant pump, at  P , generates a resonant Stokes wave, at  B , which is then up-converted to  B +f LO .  B +f LO being nearby  P , the pump locks to this frequency. (c) Cascade effect leading to the linewidth narrowing process. As the pump is optically reinjected by  B +f LO , its linewidth narrows down enhancing the Stokes wave generation process. After a few round trips the pump and the resonant Stoke waves get the same spectral purity. (d) Real time screen capture of the beatnote between the pump and the Stokes wave,  P  B , observed with an electrical spectrum analyzer. (1) and (2), the up-converted Stokes wave is not reinjected into the DFB pump laser. (2) 100 times average of (1). From (3) to(6), time evolution of the beatnote when the up-converted Stokes wave is reinjected into the DFB pump (with 100 times average). The beatnote shrinks while its power spectal density increases showing that the energy originally contained in (2) concentrates in this narrow line.

Fig. 3 :

 3 Fig. 3: Spectral purity. (a) The SAPO has been assembled in a compact thermalized housing prior to noise characterization. (b) Self-heterodyne spectra of the DFB laser outside the SAPO (left-hand side) and when it is imbedded in the SAPO (right-hand side). (c) Same as (b) with decreased spectral analysis span width: (1) 5 MHz, (2) 500 kHz and (3) 50 kHz. (d) Frequency noise computed from the phase noise of the self-heterodyne signal. The straight green line accounts for the -separation line which reveals a flicker noise linewidth of 200 Hz for 0.1 s integration time. The natural linewidth deduced from the frequency noise floor is 40 mHz.

Fig. 4 :

 4 Fig. 4: Intensity noise. Relative intensity noise (RIN) spectra of the SAPO at (a) high frequency, (b) medium frequency and (c) low frequency. The SAPO intensity noise sticks to the shot noise level of -160 dB/Hz for 5 mW photo-detected power. Below 100 kHz, the optical noise is hidden by the electronic noise of the detection chain.

Fig. 5 :

 5 Fig. 5: Frequency stability. Two identical SAPOs have been assembled in order to assess the long-term frequency stability in the free running regime. (a) Beatnote between the two independent SAPOs observed with an electrical spectrum analyzer (span width of 300 kHz). (b) Standard Allan deviation computed from the frequency drift of the beatnote (a).

Fig. 6 :

 6 Fig. 6: Evidence of self-adaptive oscillation. (a) Frequency dragging mechanism. When the Stokes wave is frequency shifted, the pump wave follows this shift and consequently the same holds for the induced Brillouin gain. This mechanism prevents any mode hopping. In (a) and (b) the SAPO relative frequency is obtained from two SAPOs beating together where one SAPO only is perturbed. (b) Experimental evidence of the frequency dragging mechanism. On one SAPO, the fiber Brillouin resonator temperature is varied. The SAPO frequency is thus dragged over 500 FSRs without any mode hop. (c) When the DFB laser current is varied, the SAPO frequency remains constant although this perturbation is equivalent to 700 FSRs of the Brillouin resonator. These two features, shown in (b) and (c), shed light on the reason why a SAPO offers such robust single frequency oscillation.

Ultra-narrow linewidth self-adaptive photonic oscillator: principle and realization M. Alouini et al.[START_REF] Yoshida | 10 channel WDM 80 Gbit/s/ch, 256 QAM bi-directional coherent transmission for a high capacity next-generation mobile fronthaul[END_REF]


C Edrick

T Bonnet

C Edric Morio

S Ebastien Szaffarczyk

Patrice R Rougier

C T Bonnet

C Morio

S Szaffarczyk

Postural Mechanisms to Control Body Displacements in the Performance of Lateral Gaze Shifts

Keywords: postural control mechanisms, postural coordination, visual tasks, medialateral axis

Medialateral postural control mechanisms (bodyweight distribution and center of pressure location) have been studied in static conditions. Our objective was to determine how these mechanisms are adjusted to perform voluntary movements, in our case 80 lateral gaze shifts at 0.125 Hz and 0.25 Hz. In healthy, young adults, we expected body marker (neck, lower back) and center of pressure displacements to be significantly greater in gaze shift conditions than in the stationary gaze condition. To explain these changes in center of pressure displacement, the amplitude contribution of both mechanisms was expected to increase significantly. All these results were found accordingly. Unexpectedly, the active contribution of the bodyweight distribution mechanism was negatively related to body marker displacements in the gaze shift conditions (ns in stationary condition). Moreover, changes in the contribution of the mechanisms were statistically weaker in effect size than changes in body displacement. However, the participants were not unstable because they performed the visual tasks as requested. We propose that the strength of medialateral postural control mechanisms may not only be strengthened to control challenging ML stance conditions but also slightly weakened to allow the performance of adequate body motions in ongoing tasks.

I n stance, postural control mechanisms have to work contin- uously to restrain postural sway [START_REF] Winter | Human balance and postural control during standing and walking[END_REF]. In the medialateral (ML) axis, [START_REF] Winter | Medial-lateral and anterior-posterior motor responses associated with center of pressure changes in quiet standing[END_REF] and [START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF] proposed the existence of a bodyweight distribution and center of pressure (COP) location mechanisms. The first mechanism is performed by loading the bodyweight under one foot and thus unloading the bodyweight under the other foot. The control may be performed at the hip level (e.g., [START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF][START_REF] Winter | Medial-lateral and anterior-posterior motor responses associated with center of pressure changes in quiet standing[END_REF]. The second mechanism is performed by inversion or eversion at the ankle level. The two mechanisms were shown to play significant and complementary roles in controlling ML COP displacement [START_REF] Bonnet | Impaired mediolateral postural control at the ankle in healthy, middleaged adults[END_REF][START_REF] Gatev | Feedforward ankle strategy of balance during quiet stance in adults[END_REF][START_REF] Lafond | Postural control mechanisms during quiet standing in patients with diabetic sensory neuropathy[END_REF][START_REF] Rougier | Relative contribution of the pressure variations under the feet and body weight distribution over both legs in the control of upright stance[END_REF][START_REF] Rougier | How spreading the forefeet apart influences upright standing control[END_REF][START_REF] Termoz | The control of upright stance in young, elderly and persons with Parkinson's disease[END_REF][START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF][START_REF] Winter | Medial-lateral and anterior-posterior motor responses associated with center of pressure changes in quiet standing[END_REF]. The bodyweight distribution mechanism (referred to as loading/unloading contribution to vertical forces) was shown to be the primary mechanism when the feet are side by side. The COP location mechanism (referred to as left and right COP change) was shown to have a secondary role in explaining the ML COP displacement, sometimes significant [START_REF] Bonnet | Impaired mediolateral postural control at the ankle in healthy, middleaged adults[END_REF][START_REF] Termoz | The control of upright stance in young, elderly and persons with Parkinson's disease[END_REF] but sometimes not significant [START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF][START_REF] Winter | Medial-lateral and anterior-posterior motor responses associated with center of pressure changes in quiet standing[END_REF]. Former studies showed that the closer the feet are, the higher the active contribution (represented by higher cross-correlations in narrow stance than in standard stance; cf. Method) of both mechanisms is (cf. [START_REF] Bonnet | Impaired mediolateral postural control at the ankle in healthy, middleaged adults[END_REF][START_REF] Gatev | Feedforward ankle strategy of balance during quiet stance in adults[END_REF] with greater changes for the COP location mechanism [START_REF] Bonnet | Impaired mediolateral postural control at the ankle in healthy, middleaged adults[END_REF]. These results have been found in static conditions (in changing the foot position) with no focal motions to be performed. However, in upright stance, individuals are rarely static; they move and perform tasks all the time [START_REF] Riccio | Affordances as constraints on the control of stance[END_REF]). Hence, these published studies may provide an initial but incomplete understanding of these mechanisms. We decided to investigate how the postural control mechanisms work in conditions requiring body motions.

A recent study by Bonnet, Cherraf, and Do (2014a) explained that Winter et al.'s (1993[START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF] model cannot be used all the time. This study showed that the anteroposterior (AP) bodyweight distribution and COP location mechanisms cannot be studied with the feet side by side. It also showed that the two mechanisms should not be studied in Tandem Romberg (with one foot forward the other). We preferred analyzing both mechanisms in the ML axis for three reasons. First, our goal was to provide representative data of everyday life and the feet side-by-side condition is more common than the Tandem Romberg condition. Second, the literature reports showed a closed link between ML postural instability and ML fall [START_REF] Maki | Prospective study of postural balance and risk of falling in an ambulatory and independent elderly population[END_REF] and between ML fall and hip fractures [START_REF] Rogers | Lateral Stability and Falls in Older People[END_REF]. Investigations to better understand the functioning of ML postural control mechanisms are thus needed. Third, this present study served to provide preliminary data to understand Parkinson's disease-related deficiencies in ML postural control.

Some studies involving ML active conditions have been published in the literature. The participants had to perform small or large ML gaze shifts to reach a single target per trial (Anastopoulos, Ziavra, [START_REF] Anastasopoulos | Gaze displacement and inter-segmental coordination during large whole body voluntary rotations[END_REF][START_REF] Hollands | A new paradigm to investigate the roles of head and eye movements in coordination of whole-body movements[END_REF][START_REF] Sklavos | Kinematic redundancy and variance of eye, head and trunk displacements during large horizontal gaze reorientations in standing humans[END_REF]. Other participants also had to reach a target appearing left and right at different frequencies and amplitudes [START_REF] Stoffregen | Postural sway and the frequency of horizontal eye movements[END_REF][START_REF] Stoffregen | Postural stabilization of visually guided eye movements[END_REF]. In all these studies, the authors analyzed COP displacement, postural displacement, and/or postural coordination. They could not analyze the contribution of the ML bodyweight distribution and COP location mechanisms because they only used a single force platform (for the need of two platforms; cf. [START_REF] Winter | Medial-lateral and anterior-posterior motor responses associated with center of pressure changes in quiet standing[END_REF]. Recently, [START_REF] Bonnet | Large lateral head movements and postural control[END_REF] used two force platforms in an active study. Twelve young adults performed two gaze shift conditions in which they had to track the appearance of a dot on the right and left at 150 of visual angle and at a frequency of 0.5 or 1 Hz. In this study, COP displacement in standard stance (feet in spontaneously chosen positions) and wide stance (feet about 40 cm apart) were significantly greater than in the stationary gaze condition. The two ML postural control mechanisms were not analyzed yet. In Bonnet and Despretz, the visual tasks engaged a great amount of behavioral variability. We decided to investigate easier visual tasks that could be performed later by patients with Parkinson's disease and their healthy older controls.

In the present study, we sought to determine how the ML bodyweight distribution and COP location mechanisms were adjusted to perform 80 lateral gaze shifts at 0.125 Hz and 0.25 Hz. We expected ML body displacements (lower back, neck) and COP displacement to be significantly greater in gaze shift conditions as a result of body (head notably) motions to perform the task (Hypothesis 1). For the control of ML stance, the contribution of the ML bodyweight distribution and COP location mechanisms was expected to be greater (Hypothesis 2) and more significantly related to ML body displacements in the gaze shift conditions than in the stationary gaze condition (Hypothesis 3). In the nonchallenging stationary gaze condition, the mechanisms were not expected to be significantly related to body displacement since the mechanisms are robust to changes [START_REF] Termoz | The control of upright stance in young, elderly and persons with Parkinson's disease[END_REF]. More significant findings were expected in the 0.25 Hz condition than in the 0.125 Hz condition (Hypothesis 4), as the former condition required twice more gaze shifts and supposedly greater postural control adjustments than the latter one.

Method

Participants

Sixteen healthy, students from the Universities of Lille participated in this study. They were included because none of them had a history of neurological or musculoskeletal disease, vestibular problems or recurrent dizziness. All gave their written, informed consent to participation and the study was performed in accordance with the tenets of the Declaration of Helsinki. The group's mean age, bodyweight and height were 21.59 § 2.65 years, 61.94 § 10.08 kg, and 1.72 § 0.10 m s, respectively.

Apparatus

A dual-top force platform (AMTI, Watertown, MA) was used to record forces and moments under each foot. The platform was placed 1.50 m from the center of a panoramic display (180 , radius: 2.1 m; height: 2.1 m; Figure 1). Its sampling frequency was set to 120 Hz. Printed lines marked the position of the heels and big toes (the subjects were barefoot) to get a stance width of 14 cm and stance angle of 17 (normative values by [START_REF] Mcilroy | Preferred placement of the feet during quiet stance: Development of a standardized foot placement for balance testing[END_REF].

A two-camera motion analysis system (Version 7.5, SIMI Reality Motion Systems GmbH, Munich, Germany) was used to record body marker displacements (marker diameter: 2.5 cm) with a sampling frequency of 15 Hz. The reflective markers were attached to the back of a hip belt (the lower back marker), of the neck (the neck marker), and of a headset (the head marker). Special lights on each of the two cameras (Led Lenser P3 8403, LED torch) were used to light the markers.

A head-mounted eye tracker (SensoMotoric Instruments, Teltow, Germany) was attached to the headset worn by the participants (Figure 2). On the computer screen, the video showed the visual environment and (as a cross) what the right eye was looking. The iViewX system recorded the pupil position at a sampling rate of 50 Hz.

A computer software projected a black dot (5 of visual angle) onto the panoramic display (Figure 1) at the participant's eye height.

All apparatus were synchronized.

Conditions

In the stationary control gaze condition, the participants had to look at the stationary dot in front of them (Figure 1). In the two gaze shift conditions, the participants had to track a dot that appeared alternately to their left and right at a visual angle of 80 (Figure 1) at 0.125 Hz or 0.25 Hz.

We decided to perform two visual tasks and not only one to test whether ML postural control could be different in a very slow visual task (the 0.125 Hz condition) and in a slow visual task (the 0.25 Hz condition). These data will also be relevant for our future studies with older adults and patients with Parkinson's disease. In the 0.125 Hz and 0.25 Hz conditions, gaze shifts had to be performed quickly once the target had disappeared but with no anticipation (cf. [START_REF] Bonnet | Large lateral head movements and postural control[END_REF]. In each condition, the participants had to relax with their arms extended along their side. During trials, the participants were aware that they had to continuously look through the small window of the eye tracker (cf. Figure 2) to perform each single gaze shift; no eye data could be recorded otherwise. Thus, they understood that their head had to rotate their gaze quickly about 80 to reach the target in the way they found it most comfortable (that is with their head or trunk or combined motions, instruction not suggested to the participants yet). There were four trials (32 s each) per visual condition to ensure adequate statistical power. The order of the conditions was randomized.

Dependent Variables

COP and body marker displacements. As in former works, the standard deviation (SD), range, and mean velocity were used to analyze COP and body marker displacements (e.g., [START_REF] Bonnet | Large lateral head movements and postural control[END_REF][START_REF] Day | Effect of vision and stance width on human body motion when standing: Implications for afferent control of lateral sway[END_REF][START_REF] Mouzat | The effect of feet position on orthostatic posture in a female sample group[END_REF]. For control purposes, the COP and body marker positions as well as the bodyweight distribution were analyzed (cf. [START_REF] Rougier | Relative contribution of the pressure variations under the feet and body weight distribution over both legs in the control of upright stance[END_REF][START_REF] Tarantola | Human stance stability improves with the repetition of the task: Effect of foot position and visual condition[END_REF].

Contribution of the ML bodyweight and COP location mechanisms.

With one single force platform, it is not possible to measure the loading/unloading of bodyweight (referred to as COP v in the model computation) and the COP displacement under each foot (reffered to as COP c , cf. [START_REF] Winter | Medial-lateral and anterior-posterior motor responses associated with center of pressure changes in quiet standing[END_REF]. For investigating the two mechanisms, we used our dual-top force platform and an updated version [START_REF] Rougier | Relative contribution of the pressure variations under the feet and body weight distribution over both legs in the control of upright stance[END_REF][START_REF] Rougier | How spreading the forefeet apart influences upright standing control[END_REF] of the validated model of ML postural control [START_REF] Lafond | Postural control mechanisms during quiet standing in patients with diabetic sensory neuropathy[END_REF][START_REF] Termoz | The control of upright stance in young, elderly and persons with Parkinson's disease[END_REF][START_REF] Winter | Medial-lateral and anterior-posterior motor responses associated with center of pressure changes in quiet standing[END_REF][START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF]:

COP net .t/ D COP l .t/ R vl .t/ R vl .t/ C R vr .t/ C COP r .t/ R vr .t/ R vl .t/ C R vr .t/ (1) COP c .t/ D COP l .t/ £ meanR vl C COP r .t/ £ meanR vr (2) COP v .t/ D meanCOP l R vl .t/ R vl .t/ C R vr .t/ C meanCOPr R vr .t/ R vl .t/ C R vr .t/ (3)
In the three equations, COP l (t) and COP r (t) are the COP displacement under the left and right foot, respectively. COP net is the resultant COP displacement under both feet. R vl (t) and R vr (t) are the vertical reaction forces under the left and right foot, respectively. MeanCOP l , meanCOP r , meanR vl , and meanR vr are the mean of each time series.

Equation 1 simply shows how to compute the COP displacement with two force platforms (cf. [START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF]. In Equation 2, the COP c displacement was calculated by eliminating the COP net displacement explained by the COP v displacement (constant mean of bodyweight under both feet throughout the trial). In Equation 3 the COP v displacement was calculated by eliminating the COP net displacement explained by the COP c displacement (constant mean of COP location under both feet throughout the trial). Figure 3 shows the result of these equations for one trial, as an example. This mathematical model is explained in former published articles (e.g., [START_REF] Rougier | Relative contribution of the pressure variations under the feet and body weight distribution over both legs in the control of upright stance[END_REF][START_REF] Rougier | How spreading the forefeet apart influences upright standing control[END_REF][START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF].

The three time series from the above mathematical model are the COP net (Equation 1), the COP c (Equation 2) and the COP v time series (Equation 3). These COP c and COP v time series then need to be analyzed to know the contribution of each mechanism to explain the COP displacement (COP net (t)). To do so, two complementary analyses were performed.

The first analysis was concerned with the amplitude contribution, or strength, of the mechanisms. We analyzed the absolute amplitude contribution represented by the variability of COP c (t) and COP v (t) with no reference to the variability of COP net (t). In our study, this analysis was useful to investigate changes in the amplitude contribution of the mechanisms from the static to the active conditions. We performed this analysis with SD (SD COP c (t) and SD COP v (t)) as in [START_REF] Rougier | Relative contribution of the pressure variations under the feet and body weight distribution over both legs in the control of upright stance[END_REF][START_REF] Rougier | How spreading the forefeet apart influences upright standing control[END_REF]. We also compared the SD of COP c and COP net time series and of COP v and COPnet time series. This analysis was called relative amplitude contribution because it is relative to COP net (t). The two variables representative of this relative amplitude contribution were %SD COP v /COP net and %SD COP v /COP net . This analysis was useful to investigate changes in the amplitude contribution of the mechanisms relative to the COP displacement to be explained.

The second analysis calculated the cross-correlation coefficient between COP c (t) and COP v (t) on the one hand and COP net (t) on the other hand (denoted as COP c vs. COP net and COP v vs. COP net , respectively; [START_REF] Gatev | Feedforward ankle strategy of balance during quiet stance in adults[END_REF][START_REF] Lafond | Postural control mechanisms during quiet standing in patients with diabetic sensory neuropathy[END_REF][START_REF] Rougier | Relative contribution of the pressure variations under the feet and body weight distribution over both legs in the control of upright stance[END_REF][START_REF] Rougier | How spreading the forefeet apart influences upright standing control[END_REF][START_REF] Termoz | The control of upright stance in young, elderly and persons with Parkinson's disease[END_REF][START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF][START_REF] Winter | Medial-lateral and anterior-posterior motor responses associated with center of pressure changes in quiet standing[END_REF]. This analysis compared the similarity of the COP c versus COP net and COP v versus COP net time series, both in terms of direction and proportionality of the time series. These analyses were not concerned with the respective amplitudes of the time-series (as in the first analysis; cf. Bonnet, Cherraf, Szaffarczyk & Rougier, 2014b). As proposed by Bonnet et al. (2014a;2014b), these cross-correlation analyses may be concerned with the degree of active contribution of the mechanism to control the COP net displacement -whether the mechanisms were sufficiently active or not. As proposed by Bonnet et al. (2014aBonnet et al. ( , 2014b)), we assumed that the higher the cross-correlation coefficient, the higher the active contribution of the postural mechanism to controlling the ML COP displacement.

For the analyses, both changes in the amplitude and active contributions were relevant to discuss the overall contribution of that mechanism. This overall contribution was calculated in taking into account both the amplitude and active contributions of each mechanism. If both amplitude and active contributions were higher or lower in one condition than in another condition, the overall contribution was assumed to be higher or lower, respectively. If both amplitude and active contribution had opposite trends (one getting higher, the other getting lower), we compared the effect sizes of each analysis to decide whether the overall contribution was higher or lower in the first condition than in the second condition. Indeed, we had no reason to assume that the amplitude contribution was more important than the active contribution, or vice versa, to discuss the overall contribution of the ML bodyweight distribution and COP location mechanisms.

Eye-in-orbit, head-on-trunk, and trunk-in-space angular displacements. The video of the eye tracker was not recorded but it could be watched online by the experimenter during trials. Hence, the experimenter could ask the participant to repeat one trial if the task was not well performed. The participants performed the task as requested if they moved the eyes from the disappearing target directly to the appearing target, thus at the expected amplitude, frequency and timing. The detail of the eye position was recorded.

The eye-in-orbit angular displacement corresponded to the angle formed by the eye relative to the orbit referential. This angle was obtained in using the ML linear displacement of the eye and the inverted tangent function. The head-in-space and trunk-in-space angular displacements corresponded to the planar yaw angles of the head-neck vector and the neck-lower back vector projected on the horizontal plane, respectively. These angular displacements were obtained from the SIMI Reality Motion System software directly. Because this system recorded data relative to the earth reference, the head-on-trunk angular displacement was obtained by subtracting the trunk-in-space angular displacement from the head-in-space angular displacement.

The time series of eye-in-orbit and head-on-trunk angular displacements (left-right turns) and stationary position (to look at the target) were clearly pictured in each single trial. An example of head-on-trunk angular displacement time series in a 0.25 Hz condition is shown in Figure 4. We calculated the mean angular positions of the eye-in-orbit and head-on-trunk when the target was on the right and left (between the vertical lines on Figure 4). The resulting left and right mean angles (two and four mean angles in the 0.125 and 0.25 Hz conditions, respectively) were averaged for each trial. The time series of trunk-in-space angular displacement were subject-dependent. These later time series showed clear transition and stationary time series in two participants, they were related to the target-or head-motion and half-time completely random in seven participants and they were completely random in seven participants. The Figure 4 shows a completely random-not related to the target position-time series of trunk-in-space angular displacement. To calculate the mean left and right angular positions of the trunk-in-space, we looked at the two and four similar time series as for the head time series (same beginning and same end) in the 0.125 Hz and 0.25 Hz conditions, respectively. In our example in Figure 4, the four left means (m1_lt, m2_lt, m3_lt, m4_lt) and the four right means (m1_rt, m2_rt, m3_rt, m4_rt) of trunk-in-space angular displacement were calculated at the same moment as the four left means (m1_lh, m2_lh, m3_lh, m4_lh) and the four right means (m1_rh, m2_rh, m3_rh, m4_rh) of head-on-trunk angular displacement.

Data Analysis

All the dependent variables (COP displacement, body and angular displacement, amplitude and active contributions of ML bodyweight and COP location mechanisms) were analyzed in the ML axis.

All the dependent variables were normally distributed and there was no outlier. Comparisons between conditions were performed with one-way repeated measures analyses of variance (ANOVAs). Pearson's correlations were used to test linear relationships between (a) the contribution of both mechanisms and (b) the body marker displacements and angular displacements in each condition. All analyses were performed at p < .01.

Results

Visual Performance: Gaze Shifts Well Performed

During the experimental session, the experimenter continuously verified that the participants kept their eyes fully on the target in the stationary gaze condition and that they reached each single target in the 0.125 and 0.25 Hz conditions. As in [START_REF] Bonnet | Large lateral head movements and postural control[END_REF], no trial was repeated and the targets were easily reached at the requested amplitude, frequency and timing.

Effects of Conditions on Eye-in-Orbit, Head-on-Trunk, and Trunk-in-Space Angular Displacements

The eye-in-orbit angular displacement was significantly greater in the 0.125 Hz (10.17 § 5.61 ) than in the 0.25 Hz conditions (8.55 § 5.59 ), F(1, 15) D 10.89, p < .05. The eyes almost did not move in the stationary gaze condition (1.35 § 0.52 ).

The head-on-trunk and trunk-in-space angular displacements were similar between the two gaze shift conditions, F s (1, 15) < 1.15, p >.05). On average in both tasks, the mean head-on-trunk and trunk-in-space angular displacements were 58.44 § 10.58 and 12.20 § 10.07 , respectively. These values were greater than the ones in the stationary gaze condition (0.25 § 0.65 , 0.21 § 0.64 ).

Overall therefore, healthy, young adults performed 80 gaze shifts mostly in turning their head (almost three quarters of body angular displacement) and additionally their eyes and trunk.

Effects of Conditions on COP and Body Linear Displacements and on the Contribution of the ML Bodyweight and COP Location Mechanisms

The results of the ANOVA are reported in Table 1. Overall, the results showed that the participants exhibited significantly higher amplitude and velocity of body and COP displacement and higher absolute amplitude contribution of the ML bodyweight and COP location mechanisms in the active conditions than in the stationary gaze condition.

Contributions of the Mechanisms Significantly Related to Body Displacements?

The results of the Pearson's correlation analyses are reported in Table 2. These analyses showed that there were more significant relationships between body displacements and the contributions of the two mechanisms in the active gaze shift conditions than in the stationary gaze condition.

Complementary Analyses

Figures of the time series.

A close look at the figures of the COP, back, trunk, and head time series showed that all these data were random-like in the stationary gaze conditions. For all the participants in the 0.125 and 0.25 Hz conditions, the neck time series mostly showed a square wave displacement of the marker (as the head-on-trunk displacement in Figure 4). Only for half of the participants (8 of 16) in the 0.125 and 0.25 Hz conditions, the back time series showed a square wave displacement of the marker much less clearly pictured than the neck time series yet. The other half time series showed random-like displacement, that is with no square wave displacement-as the trunk-in-space displacement in Figure 4. The COP displacement was random-like in all trials, even in the 0.125 and 0.25 Hz conditions. Note. The variables of angular body displacement (eyes, head, trunk) were also entered in the analyses but did not provide any significant relationship. Thus, they are not reported in the table. See Table 1 for the definition of these terms. All rs are significant at p < .01. COP D center of pressure; SD D standard deviation.

Asymmetry. The analyses did not show any significant effect in the ML COP mean position, body marker mean position and load/unload weight repartition under each foot, F s (2, 30) < 1.21, p >.05.

Controlling for confounding variables. We used the detrending normalization procedure recommended by [START_REF] O'malley | Normalization of temporal-distance parameters in pediatric gait[END_REF] to remove the influence of age, height, and weight on the calculated time series. We performed again the ANOVAs on the normalized data. These analyses did not change any significant finding. They only slightly strengthened some results (n 2 p increased by .01-.03), mostly when controlling for weight. Hence, the participants' physical characteristics were not confounding variables.

Discussion

To perform the gaze shifts, the participants exhibited greater body marker displacements and greater COP displacements. Consistent with expectation, the two mechanisms contributed more to control ML body displacements in the gaze shift conditions than in the stationary gaze condition. However, the contribution of the bodyweight distribution mechanism tended to be weaker than expected, in relative terms, in gaze shift conditions. As we discuss, these results may show how ML postural control was adjusted to control ML stance while allowing body motions to perform the requested gaze shifts.

The Tasks Were Well Performed

In past published articles, visual performances in gaze shift conditions were always performed as requested (e.g., [START_REF] Anastasopoulos | Gaze displacement and inter-segmental coordination during large whole body voluntary rotations[END_REF][START_REF] Bonnet | Large lateral head movements and postural control[END_REF][START_REF] Rey | Saccades horizontal or vertical at near or at far do not deteriorate postural control[END_REF][START_REF] Sklavos | Kinematic redundancy and variance of eye, head and trunk displacements during large horizontal gaze reorientations in standing humans[END_REF][START_REF] Stoffregen | Postural sway and the frequency of horizontal eye movements[END_REF][START_REF] Stoffregen | Postural sway and the frequency of horizontal eye movements[END_REF]. In the present study, the participants also performed the visual tasks, as requested in terms of amplitude, frequency, and timing. Thus, the participants organized their postural control to perform the task without losing their equilibrium. In the literature, some studies already analyzed postural coordination (eyes, head, lower back, feet movement) under large gaze shift conditions [START_REF] Hollands | A new paradigm to investigate the roles of head and eye movements in coordination of whole-body movements[END_REF]Anastopoulos et al., 2009;[START_REF] Sklavos | Kinematic redundancy and variance of eye, head and trunk displacements during large horizontal gaze reorientations in standing humans[END_REF]. These studies with healthy, young adults showed that postural coordination was organized to facilitate gaze and head motions to reach the targets. Additionally, the higher the body part, the quicker the movement onset (Anastopoulos et al., 2009;[START_REF] Hollands | A new paradigm to investigate the roles of head and eye movements in coordination of whole-body movements[END_REF]. The eyes vs. head onset latency and the eyes versus foot amplitude displacement exhibited the greatest correlation coefficient [START_REF] Hollands | A new paradigm to investigate the roles of head and eye movements in coordination of whole-body movements[END_REF]. We additionally showed that lower back and neck ML displacements were significantly greater in range and standard deviation in the active gaze shift conditions compared with the stationary gaze condition. As in [START_REF] Bonnet | Large lateral head movements and postural control[END_REF], the magnitude (range, SD, velocity) of COP displacement was significantly greater in the gaze shift conditions. This last finding was expected because the COMor body-displacement is known to be controlled by the COP displacement [START_REF] Winter | Human balance and postural control during standing and walking[END_REF]. The body displacement was greater in gaze shift conditions and the COP displacement necessarily needed to be greater. Thus, as in Bonnet and Despretz, the participants were actively stable in gaze shift conditions allowing greater magnitude of body displacements to perform the gaze shifts. These results were consistent with Hypothesis 1.

Several results showed that the 0.25 Hz condition was not clearly more challenging than the 0.125 Hz condition in healthy, young adults. Hence, Hypothesis 4 was not confirmed. Only one significant relationship between SD COP v and SD of the lower back displacement in the 0.25 Hz condition could not be found in the 0.125 Hz condition. The eye-in-orbit angular displacement was significantly lower in the 0.25 Hz than in the 0.125 Hz condition but the headon-trunk and trunk-in-space angular displacements were similar between these conditions. Hence, the heavy head and trunk segments coordinated similarly in both 0.25 Hz and 0.125 Hz conditions leaving only differences in no heavy body parts (the eyes). Moreover, the neck and lower back markers moved significantly quicker in the 0.25 Hz condition than in the 0.125 Hz condition (Table 1), most probably as a consequence of our method. Indeed, the figures of the data (cf. complementary analyses) showed twice more square wave like displacements of the markers in the 0.25 Hz condition than in the 0.125 Hz condition. These additional displacements necessarily increased the mean velocity of the markers. These displacements had no incidence on the magnitude (R and SD) of COP, neck and lower back displacements, thus showing that the variability of ML body displacement was controlled similarly in both conditions. One should note that the postural control signal was more and more random-like from the top (eyes) to the bottom of the body (COP). In the rest of the present discussion, the results of both conditions will be combined. We now discuss how the two postural control mechanisms were adjusted to perform the visual tasks.

Changes in the Contribution of the ML Bodyweight Distribution and COP Location Mechanisms

The absolute amplitude contribution of both ML bodyweight distribution and COP location mechanisms was significantly greater in active gaze shift conditions than in the stationary gaze condition (Table 1). Hypothesis 2 was thus validated. We expected to find so to explain greater amplitude of COP displacement in gaze shift conditions. However, the relative amplitude contribution did not change significantly (Table 1). In other words, the increase in the amplitude contribution of both ML bodyweight distribution and COP location mechanisms were only proportional-not significantly greater or smaller-to the increase in COP displacement. The strength of postural control mechanisms was minimally adapted to the active task performed. By contrast in the literature of static studies, the relative amplitude contribution of both ML bodyweight distribution and COP location mechanisms was modified by changing the stance width. In [START_REF] Bonnet | Impaired mediolateral postural control at the ankle in healthy, middleaged adults[END_REF], it was found that the closer the feet, the lower the relative amplitude contributions of both mechanisms, probably because the less effective the force generated to control ML postural displacement [START_REF] Henry | Effect of stance width on multidirectional postural responses[END_REF][START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF].

In the present active study, we found approximately equal changes in the ML bodyweight distribution and COP location mechanisms. On the one hand, the significant change in amplitude contribution was slightly greater for the bodyweight distribution mechanism than for the COP location mechanism between the stationary and gaze shift conditions (n 2 p in Table 1). On the other hand, the relationships between COP c and body motions were stronger in gaze shift conditions while no such result was found for the bodyweight distribution mechanism. Surprisingly therefore, when ML postural control was actively challenged, the primary ML bodyweight distribution mechanism did not seem to change more than the secondary ML COP location mechanism to avoid losing equilibrium. One insight is that this prime mechanism may play another role than simply restraining ML body displacement, as we will suggest subsequently.

The active contribution of both ML bodyweight distribution and COP location mechanisms did not change between stationary and active conditions (no main effect in the ANOVA). Unexpectedly, significant negative relationships were found between COP v versus COP net and body marker displacements (lower back and neck) in the two gaze shift conditions (Table 2). Additionally, overall changes in COP v and COP c between conditions were weaker than changes in COP displacement, which were weaker than changes in body marker displacements (see n 2 p in Table 1). These results were inconsistent with hypothesis 3. They were unexpected because changes in the controller were expected to be stronger than changes in the controlled variable to allow-by definition-adequate postural control [START_REF] Maurer | A new interpretation of spontaneous sway measures based on a simple model of human postural control[END_REF]. These results may be key findings. Here in these active conditions, it may be that the controllers were slightly released (relative to a perfect control) to allow body motion to be performed in the visual tasks. We propose that if the contribution of the bodyweight distribution mechanism had been too high, individuals may have been too tensed to perform adequate body motions. In other words, once ML postural control was adjusted to objectively control the increase in body displacement in active conditions, slight release of the primary mechanism (pictured in the relative contribution of the bodyweight distribution mechanism) could be allowed to perform head and body motions. This a posterior interpretation does not seem to exist in the literature for one simple reason. Usually, researchers record and illustrate electromyographic (EMG) or other data in absolute terms but not in relative terms. Published studies have shown an increase in EMG activations once body motions have to be performed (e.g., [START_REF] Yiou | Adaptability of anticipatory postural adjustments associated with voluntary movement[END_REF] or when postural control has to be maintained under some kind of perturbations (e.g., [START_REF] Henry | Effect of stance width on multidirectional postural responses[END_REF]. However, these studies do not report differences between expected and measured EMG activations or other data. The possibility to analyze the absolute and relative amplitude contributions of the two mechanisms constitutes a real advantage of [START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF]1993) model to understand how these mechanisms work. In brief, our findings may show that the functioning of the ML postural control mechanisms is more complex in active than in static conditions.

Perspectives and Conclusion

A limitation of the present study is that the head, neck, and lower back rotations were not measured individually and thus not studied. Instead, we only get the head-in-space and trunk-in-space angular displacements with the SIMI Reality Motion System software. Also, the precision would have been better if we had used two markers at each level (head, shoulder, hip levels). However, our results were consistent with the condition requirements and thus validated our method. Indeed, the data showed that the participants moved their gaze (eyesChead-on-trunkCtrunk-in-space angular displacements) 80.86 and 82.37 in the 0.25 and 0.125 Hz conditions, which is within the condition requirement (80 § 2.5 on each side as the target covered 5 of visual angle at their center).

In this study, we performed easier gaze shift conditions than in [START_REF] Bonnet | Large lateral head movements and postural control[END_REF] to obtain preliminary data for a future study with older adults (healthy and affected by Parkinson's disease). This future study will be relevant to better understand the nature of age-and diseaserelated deficiencies in ML postural control. Indeed, [START_REF] Termoz | The control of upright stance in young, elderly and persons with Parkinson's disease[END_REF] already showed that postural control mechanisms are not clearly different between PD patients and their controls in static quiet stance while PD patients exhibit deficiencies in ML postural control in more challenging conditions (e.g., platform motion; cf. [START_REF] Adkin | Trunk sway measurements during stance and gait tasks in Parkinson's disease[END_REF][START_REF] Horak | Direction-specific postural instability in subjects with Parkinson's disease[END_REF].

FIGURE 1 .

 1 FIGURE 1. Description of the experimental setting (not drawn to scale). The participant stood on a dual-top force platform in the experimental room at a distance of 1.50 m from a semicircular, panoramic display. The target (a black dot, subtending a visual angle of 5 ) was presented either in a stationary position in front of the participant or alternately on the left and right at a visual angle of 80 . The dot's position was recorded with a computer program and projected onto the screen by three cameras.

FIGURE 2 .

 2 FIGURE 2. Picture of the head mounted eye-tracker that we used.

  FIGURE 3. Time series for COPnet, and COPc in one trial (30 s long) performed by the experimenter (units: cm) in the medialateral (ML) axis. The COPnet is the integrated displacement of the COP under both feet. The COPv (t) is the component of the COPnet(t) that can be explained by the bodyweight mechanism. The COPc(t) is the component of the COPnet(t) that can be explained by the center of pressure location mechanism.

FIGURE 4 .

 4 FIGURE 4. Time series for the head-on-trunk angular displacement and trunk-in-space angular displacement (units: deg D degrees) in one trial (32 s long) performed by one participant in the 0.25 Hz condition. The left and right means position of the head-on-trunk angular displacement were calculated with the four mean angular positions when the head had finished to turn to reach the left and right targets, respectively. The eight mean angular positions of the trunk-in-space angular displacement were calculated at the same moment that the eight mean angular positions of the head-in-space angular displacement. For the abbreviations: m1_lh D mean 1 when the head was on the left; m2_rh D mean 2 when the head was on the right; m1_lt D mean 1 when the trunk was on the left; m3_lt D mean 3 when the trunk was on the right.

  Time series for COPnet, and COPc in one trial (30 s long) performed by the experimenter (units: cm) in the medialateral (ML) axis. The COPnet is the integrated displacement of the COP under both feet. The COPv (t) is the component of the COPnet(t) that can be explained by the bodyweight mechanism. The COPc(t) is the component of the COPnet(t) that can be explained by the center of pressure location mechanism.

TABLE 1 .

 1 Results of the Repeated Measures ANOVA and Post Hoc Newman-Keuls Analyses D mean velocity. SD amplitudes of the COP v (COP vertical) and COP c (COP change) time series either calculated individually or expressed in percentage of the SD amplitude of the COP net (resultant displacement of the COP under both feet). SD COP v and SD COP c are representative of the absolute amplitude contribution of the bodyweight distribution and COP location mechanisms.%SD COP v /COP net and%SD COP c /COP net is representative of the relative amplitude contribution of these two mechanisms (see text). The table also shows cross-correlation coefficient in two analyses: COP v versus COP net and COP c versus COP net . These analyses are representative of the active contribution of both mechanisms (see text). The table shows averages ( § SD) in the 0.25 Hz and 0.125 Hz gaze shift conditions and in the stationary gaze condition.

		0.25 Hz	0.125 Hz	Stationary	ANOVA
	R of COP displacement (cm)	1.92 ( § 1.01) a	1.84 ( § 0.93) a	1.10 ( § 0.48)	F(2, 30) D 11.80, n 2 p D .31, p < .01
	SD of COP displacement (cm)	0.37 ( § 0.22) a	0.40 ( § 0.25) a	0.22 ( § 0.11)	F(2, 30) D 8.47, n 2 p D .27, p < .01
	V of COP displacement (cm.s ¡1 )	1.80 ( § 0.23) a	1.74 ( § 0.25) a	0.66 ( § 0.26)	F(2, 30) D 12.88, n 2 p D .32, p < .01
	R of neck displacement (cm)	4.32 ( § 1.52) a	4.40 ( § 1.87) a	1.30 ( § 0.57)	F(2, 30) D 49.10, n 2 p D .43, p < .01
	SD of neck displacement (cm)	1.21 ( § 0.44) a	1.27 ( § 0.56) a	0.31 ( § 0.15)	F(2, 30) D 52.24, n 2 p D .44, p < .01
	V of neck displacement (cm.s ¡1 )	0.93 ( § 0.29) ab	0.62 ( § 0.22) a	0.28 ( § 0.11)	F(2, 30) D 88.11, n 2 p D .46, p < .01
	R of trunk displacement (cm)	2.51 ( § 1.56) a	2.71 ( § 2.00) a	0.91 ( § 0.41)	F(2, 30) D 12.87, n 2 p D .32, p < .01
	SD of trunk displacement (cm)	0.62 ( § 0.48) a	0.70 ( § 0.62) a	0.21 ( § 0.11)	F(2, 30) D 9.65, n 2 p D .28, p < .01
	V of trunk displacement (cm.s ¡1 )	0.56 ( § 0.29) ab	0.41 ( § 0.22) a	0.22 ( § 0.09)	F(2, 30) D 19.10, n 2 p D .36, p < .01
	SD COP v	0.35 ( § 0.16) a	0.37 ( § 0.20) a	0.21 ( § 0.10)	F(2, 30) D 9.41, n 2 p D .28, p < .01
	SD COP c	0.15 ( § 0.13) a	0.15 ( § 0.16) a	0.06 ( § 0.03)	F(2, 30) D 6.38, n 2 p D .23, p < .01
	%SD COP v /COP net	0.97 ( § 0.15)	0.97 ( § 0.15)	0.97 ( § 0.07)	F(2, 30) D 0.02, p > .01
	%SD COP c /COP net	0.35 ( § 0.15)	0.33 ( § 0.18)	0.26 ( § 0.08)	F(2, 30) D 2.59, p > .01
	COP v vs. COP net	0.93 ( § 0.05)	0.93 ( § 0.07)	0.96 ( § 0.04)	F(2, 30) D 1.56, p > .01
	COP c vs. COP net	0.23 ( § 0.29)	0.25 ( § 0.36)	0.22 ( § 0.25)	F(2, 30) D 0.04, p > .01

Note. COP D center of pressure; R D range; SD D standard deviation; V a Significant difference between a gaze shift condition and the stationary gaze condition. b Significant difference between the two gaze shift conditions.

TABLE 2 .

 2 Results of the Pearson's Correlation Analyses (r) Between Variables of Medialateral Postural Control Mechanisms (SD COP v , SD COP c , COP v vs. COP net , COP c vs. COP net ) and Variables of Linear Body SD Displacement (Head, Neck, Trunk) in the 0.25 Hz, 0.125 Hz, and Stationary Conditions

			0.25 Hz		0.125 Hz	Stationary
		Trunk SD Neck SD Head SD Trunk SD Neck SD Head SD Trunk SD Neck SD Head SD
	COP v versus COP net	¡.78	¡.68	¡.86	¡.73
	COP c versus COP net
	SD COP v	.81	.70	.79	.97	.90	.89
	SD COP c	.76	.68	.83	.64
	%SD COP v /COP net
	%SD COP c /COP net	.89	.76	.88	.75

Journal of Motor Behavior

2015, Vol. 47, No. 1

Received October 2, 2013 Revised May 8, 2013 Accepted May 11, 2014 2015, Vol. 47, No. 1