A Gusching

M Petersen

N Passilly

D Brazhnikov

M Abdel Hafiz

R Boudot

Short-term stability of Cs microcell-stabilized lasers using dual-frequency sub-Doppler spectroscopy

The combination of atomic spectroscopy, integrated photonics and microelectromechanical systems (MEMS) paves the road to the demonstration of microcell-based optical atomic clocks. Here, we report the short-term stability budget of table-top Cs microcell-stabilized lasers based on dual-frequency sub-Doppler spectroscopy (DFSDS). The dependence of the sub-Doppler resonance properties on key experimental parameters is studied. The detection noise budget and absolute phase noise measurements are in good agreement with the measured short-term frequency stability of the laser beatnote, at the level of 1.1 × 10 -12 τ -1/2 until 100 s, currently limited by the intermodulation effect from a distributed-feedback laser setup. The fractional frequency stability of the laser beatnote at 1 s is about 100 times better than those of commercial microwave chip-scale atomic clocks and validate the interest of the DFSDS approach for the development of high-performance microcell-based optical standards.

INTRODUCTION

The interrogation of an alkali vapor atom ensemble in a mmscale cell has allowed the development of a wide variety of high-precision chip-scale atomic devices [START_REF] Kitching | Chip-scale atomic devices[END_REF]. Among these instruments, miniaturized microwave atomic clocks based on coherent population trapping (CPT) [START_REF] Vanier | Atomic clocks based on coherent population trapping: a review[END_REF] have known a remarkable development and progress, including their commercialization [3], due to their unrivaled size-power-frequency stability budget [START_REF] Lutwak | The Chip-Scale Atomic Clock-Prototype Evaluation[END_REF][START_REF] Zhang | ULPAC: A Miniaturized ultralow-power atomic clock[END_REF][START_REF] Vicarini | Mitigation of temperature-induced light-shift effects in miniaturized atomic clocks[END_REF][START_REF] Yanagimachi | Reducing frequency drift caused by light shift in coherent population trapping-based low-power atomic clocks[END_REF]. These clocks are now widely-used in numerous applications, including underwater sensor networks, secure communications or satellite-based navigation systems. However, microwave CSACs suffer from drawbacks and limitations. These clocks use in general vertical-cavity surfaceemitting lasers (VCSELs) whose frequency noise and variations can limit the clock frequency stability [START_REF] Yanagimachi | Reducing frequency drift caused by light shift in coherent population trapping-based low-power atomic clocks[END_REF][START_REF] Zhang | Rubidium chipscale atomic clock with improved long-term stability through light intensity optimization and compensation for laser frequency detuning[END_REF]. In addition, the presence of buffer gas pressure in the cell induces an important frequency shift of the clock transition [START_REF] Kozlova | Temperature and pressure shift of the Cs clock transition in the presence of buffer gases: Ne, N 2 , Ar[END_REF] that can limit the clock mid-term frequency stability and jeopardize the establishment of a clock accuracy budget. Over the last decade, state-of-the-art microwave atomic clocks (i.e. fountains [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF]) have been outperformed by optical atomic clocks that rely on the frequency stabilization of a low noise laser onto a high quality-factor optical atomic transition [START_REF] Schioppo | Ultrastable optical clock with two cold-atom ensembles[END_REF][START_REF] Grotti | Geodesy and metrology with a transportable optical clock[END_REF][START_REF] Mcgrew | Atomic clock performance enabling geodesy below the centimetre level[END_REF][START_REF] Wcislo | New bounds on dark matter coupling from a global network of optical atomic clocks[END_REF][START_REF] Sanner | Optical clock comparison for Lorentz symmetry testing[END_REF][START_REF] Oelker | Demonstration of 4.8 × 10 -17 stability at 1 s for two independent optical clocks[END_REF]. However, these instruments, making use of laser cooling, lattice-confined atoms or trapped ions, remain complex and their deployment outside the laboratory is challenging. To satisfy applications with stringent SWaP (size-weight-power) specifications, an attractive approach concerns the development of high-performance vapor cell-based optical clocks using Dopplerfree interrogation schemes [START_REF] Schuldt | Optical clock technologies for global navigation satellite systems[END_REF]. In this domain, the two-photon transition at 778 nm in Rb vapor is an attractive candidate due to its narrow natural linewidth of about 300 kHz [START_REF] Touahri | Frequency measurement of the 5S1/2 (F=3)-5D5/2 (F=5) two-photon transition in rubidium[END_REF][START_REF] Martin | Compact optical atomic clock based on a two-photon transition in rubidium[END_REF][START_REF] Martin | Frequency shifts due to Stark effects on a rubidium two-photon transition[END_REF]. This approach was recently used for the demonstration of a microcell-based optical clock with remarkable stability performances at the level of 4 × 10 -12 τ -1/2 until 1000 s [START_REF] Newman | Architecture for the photonic integration of an optical atomic clock[END_REF], later improved to 2.9 × 10 -12 τ -1/2 until 100 s using a micro-optics breadboard [START_REF] Maurice | Miniaturized optical frequency reference for next-generation portable optical clocks[END_REF]. More recently, using a low noise external cavity diode laser (ECDL), the same group at NIST reported a Rb-microcell optical frequency standard with an exceptional frequency stability of 1.8 × 10 -13 τ -1/2 until 100 s [START_REF] Newman | Highperformance compact optical standard[END_REF]. The development of high-performance cell optical clocks based on the alternative 780-776 nm two-photon transition in Rb was also reported [START_REF] Gerginov | Two-photon optical frequency reference with active ac Stark shift cancellation[END_REF][START_REF] Perrella | Dichroic two-photon rubidium frequency standard[END_REF] but, to our knowledge, no demonstration of this approach has been performed with a MEMS cell yet. An alternative approach to the two-photon transition is the well-known sub-Doppler spectroscopy (SDS) scheme [START_REF] Haroche | Theory of saturated-absorption line shapes[END_REF][START_REF] Wieman | Doppler-free laser polarization spectroscopy[END_REF]. The latter has been widely-used for laser frequency stabilization with glass-blown vapor cells [START_REF] Rovera | A laser diode system stabilized on the cesium D 2 line[END_REF][START_REF] Affolderbach | Tuneable, stabilised diode lasers for compact atomic frequency standards and precision wavelength references[END_REF][START_REF] Liu | A Distributed-feedback diode laser frequency stabilized on Doppler-Free Cs D 1 line[END_REF][START_REF] Liang | Compact stabilized semiconductor laser for frequency metrology[END_REF][START_REF] Liang | Ultralow noise miniature external cavity semiconductor laser[END_REF][START_REF] Strangfeld | Prototype of a compact rubidium-based optical frequency reference for operation on nanosatellites[END_REF]. The use of SDS for the development of microcell-based frequency-stabilized lasers was investigated initially in [START_REF] Knappe | Microfabricated saturated absorption laser spectrometer[END_REF] with the demonstration of a micro-fabricated saturated absorption laser spectrometer and in further studies [START_REF] Gruet | A miniature frequency stabilized VCSEL system emitting at 795 nm based on LTCC modules[END_REF][START_REF] Loh | Microresonator Brillouin laser stabilization using a microfabricated rubidium cell[END_REF][START_REF] Hummon | Photonic chip for laser stabilization to an atomic vapor with 10 -11 instability[END_REF]. In [START_REF] Hummon | Photonic chip for laser stabilization to an atomic vapor with 10 -11 instability[END_REF], the frequency stabilization of a DBR laser onto a Rb microcell, using some light routing through an integrated silicon nitride waveguide and grating system to the cell, was demonstrated at the level of 10 with a microresonator comb for direct-comb spectroscopy [START_REF] Stern | Direct Kerr frequency comb atomic spectroscopy and stabilization[END_REF] or an ultra-compact Fabry-Perot cavity [START_REF] Zhang | Ultranarrow linewidth photonic-atomic laser[END_REF]. In [START_REF] Hafiz | Doppler-free spectroscopy on the Cs D1 line with a dual-frequency laser[END_REF][START_REF] Hafiz | High-contrast sub-Doppler absorption spikes in a hot atomic vapor cell exposed to a dual-frequency laser field[END_REF], the detection of high-contrast sign-reversed naturallinewidth sub-Doppler resonances, explained in a detailed quantitative model [START_REF] Brazhnikov | Dual-frequency sub-Doppler spectroscopy: Extended theoretical model and microcell-based experiments[END_REF], was reported using a dual-frequency sub-Doppler spectroscopy (DFSDS) technique. However, no detailed resonance spectroscopy and short-term stability budget were performed.

In the present paper, we report a detailed short-term stability budget of table-top Cs microcell-stabilized lasers based on DFSDS. Two similar systems inside the same thermal enclosure are compared. The first setup uses a distributed feedback (DFB) laser while the second setup uses an external-cavity diode laser (ECDL). Spectroscopy of the sub-Doppler resonance as a function of some key experimental parameters (laser power, cell temperature and magnetic field) was performed first in order to optimize the lasers' short-term frequency stability. A detailed noise analysis is then reported for each setup and found to be in good agreement with the measured short-term stability of 1.1 × 10 -12 τ -1/2 up to 100 s and absolute phase noise performances.

EXPERIMENTAL SETUP

Figure 1 shows the experimental setup. Two lasers (one distributed feedback (DFB, Eagleyard EYP-DFB-0895 model) laser and one external cavity diode laser (ECDL, Toptica DL Pro) are each frequency-stabilized onto a Cs microcell using DFSDS. The first laser system uses a DFB laser diode tuned on the Cs D 1 line at 895 nm followed by a 70 dB optical isolation stage. The light is fiber-coupled into a Mach-Zehnder electro-optic modulator (EOM, iXblue NIR-MX800-LN-10) driven by a microwave frequency synthesizer at 4.596 GHz in order to produce two first-order optical sidebands split by 9.192 GHz. The light is then sent into an acousto-optic modulator (AOM) driven at 110 MHz by a RF synthesizer. The AOM can be used for laser power control and stabilization [START_REF] Tricot | Power stabilization of a diode laser with an acousto-optical modulator[END_REF]. The +1-order is then directed into an evacuated Cs vapor microfabricated cell containing a pill Cs dispenser [START_REF] Vicarini | Demonstration of the mass-producible feature of a Cs vapor microcell technology for miniature atomic clocks[END_REF]. The laser beam diameter at the cell input is about 2 mm. The cell is made of silicon cavities etched by deep reactive ion etching (DRIE) and sandwiched between two anodically-bonded borosilicate glass wafers. During the cell fabrication, laser-activation of the pill dispenser is performed gradually, i.e. with successive steps, in order to avoid excess of alkali in the cell. Figure 1(b) shows a photograph of the microfabricated cell. Note that, over a period of several months, at a cell temperature of 62 • C used in clock operation, we did not observe any significant appearance of Cs droplets on the cell windows. The MEMS cell is held inside the physics package shown in Fig. 1(c), which is surrounded by a mu-metal magnetic shield. After the cell, the light impinges a quarter-wave plate (QWP) and is reflected back by a mirror positioned to enhance the constructive contribution of hyperfine Zeeman effects and increase the sub-Doppler resonance contrast [START_REF] Brazhnikov | Dual-frequency sub-Doppler spectroscopy: Extended theoretical model and microcell-based experiments[END_REF]. The reflected light beam is then directed using a cube right before the cell to a photodiode for the detection of the spectroscopic signal and the generation of a derivative error signal using a lock-in amplifier (Stanford Research SR-860). The latter is then processed in a proportional-integral (PI) controller to provide a correction signal that is applied to the DFB laser current. The second laser system is based on an ECDL. Similar EOM and AOM references are used, except that the AOM of the ECDL setup is driven at 80 MHz (instead of 110 MHz for the DFB setup). This allows to directly produce a 30 MHz beat-note between the two laser setups. Before the EOM, on both setups, a part of the laser beam is separated from the main beam and fiber coupled. The two laser beams are then combined into one single fiber and the resulting beat-note signal is measured by a fast photo-diode. The output signal can be analyzed with a frequency counter or a phase noise analyzer. The counter and both microwave frequency synthesizers of the experiment are all referenced to an active hydrogen maser. the sub-Doppler resonance is 126 mV for the DFB setup and 138 mV for the ECDL setup. The FWHM of the sub-Doppler resonance is 14.7 MHz and 14.2 MHz for the DFB and ECDL setup, respectively. Spectroscopy of these sub-Doppler resonances was performed to evaluate the impact of key experimental parameters, mainly the laser power, the cell temperature and the static magnetic field. Figure 3 depicts the signal height S, the FWHM and the S/FWHM ratio versus the laser power measured right in front of the cell for both setups. In the DFB setup case, no data points were recorded for laser power higher than 600 µW, due to limited power at the output of the AOM. We observe a clear increase of the resonance signal and FWHM with increased laser power. The slope of the S/FWHM ratio dependence to laser power is reduced for power values higher than about 300 µW. We did a similar study with the cell temperature, with results shown in Fig. 4. In both setups, the signal is maximized at a temperature of about 65-67 • C, while the FWHM reaches a B experienced by the atoms by changing the current applied to the Helmholtz coils (see Fig. 1(c)). Calibration of the magnetic field-versus-current dependence was performed by replacing the Cs cell by a buffer-gas filled Cs-Ne microcell, and the reflexion mirror by a photodiode to detect laser power at the direct output of the cell and scanning the microwave frequency to detect microwave CPT resonances. The application of the static magnetic field raises the Zeeman degeneracy, from which the actual applied magnetic field was extracted by measuring the frequency change of the magnetic-field sensitive hyperfine Zeeman transition between both m F = +1 ECDL Fig. 6. Detection noise at the photodiode output for both setups (DFB and ECDL). Measurements are performed with a phase noise analyzer (Rohde-Schwarz FSWP). Black curves represent the detection noise with the photodiode in the dark. Blue curves depict an image of the laser amplitude (AM) noise, with activation of the laser power servo. For this purpose, the laser frequency is tuned out of optical resonance and the laser power impinging the photodiode is reduced with a neutral density filter to be comparable to the incident one in clock configuration. Green curves yield an image of the free-running laser frequency noise (FM). Light green curves correspond to the total detection noise at the photodiode output in locked laser frequency configuration. quantum number) with the applied current. The sub-Doppler resonance is narrowed with increased magnetic field. This narrowing effect, predicted in [START_REF] Hafiz | High-contrast sub-Doppler absorption spikes in a hot atomic vapor cell exposed to a dual-frequency laser field[END_REF] (fig. 20), is explained by the increase of the Doppler profile height with higher values of B, in conjunction with the fact that, under low magnetic field values, the absorption value at the center of the resonance does not change. Note also that the signal/FWHM ratio of sub-Doppler resonances is maximized as expected at null magnetic field for both setups [START_REF] Hafiz | High-contrast sub-Doppler absorption spikes in a hot atomic vapor cell exposed to a dual-frequency laser field[END_REF].

EXPERIMENTAL RESULTS

A. Spectroscopy of the sub-Doppler resonance

B. Noise budget and absolute phase noise

Main noise source contributions to the short-term frequency stability of each laser setup are the photon shot noise, the photodetector noise (in the dark), the laser amplitude (AM) noise, the laser frequency noise through the FM-AM conversion process and the intermodulation effect due to the intrinsic phase noise (frequency noise) of the free-running laser. For a first estimation, we only consider white-noise sources. The shot-noise contribution σ sn is given by [START_REF] Yun | High-Performance coherent population trapping clock with polarization modulation[END_REF]:

σ sn (τ) = ∆ν ν 0 2 2hν 0 C 2 P o 1 τ (1)
with ∆ν being the sub-Doppler resonance linewidth (FWHM), h the Planck constant, ν 0 the laser frequency (3.35 × 10 14 Hz for the Cs D 1 line), P o the optical power incident on the photodiode and C the contrast of the resonance, defined as the ratio between the sub-Doppler resonance signal height S and the dc background level in the bottom of the sub-Doppler resonance. The contribution of the photodetector noise σ pd is given by: with σ 2 D being the power spectral density (PSD) of voltage noise at the output of the photodiode in the dark measured at the offset frequency f = f M , where f M is the laser modulation frequency, while S l = S/∆ν is the sub-Doppler resonance signal/linewidth ratio. The contribution of laser AM and laser FM noise is extracted in a similar way by measuring the PSD of voltage noise at f M in respective conditions where the laser frequency is out of optical resonance or at half-height of the sub-Doppler resonance profile, respectively. The free-running laser phase noise can limit the final laser shortterm frequency stability through the intermodulation effect [START_REF] Audoin | A limit to the frequency stability of passive frequency standards due to an intermodulation effect[END_REF], with the contribution:

σ (τ) = 1 ν 2 0 σ 2 D S 2 l 1 τ (2)
σ int (τ) f M ν 0 S ϕ (2 f M ) 1 τ (3)
with S ϕ (2 f M ) the PSD of the laser phase noise fluctuations in rad 2 /Hz measured at f = 2 f M . The detection noise can be also degraded by the laser amplitude (AM) noise and the intrinsic laser frequency noise (FM). Their contributions, σ AM-AM and σ FM-AM , are respectively obtained by:

σ AM-AM (τ) ( 1 ν 0 ) 2 σ 2 P (∆V/∆P) 2 S 2 l 1 τ (4) σ FM-AM (τ) ( 1 ν 0 ) 2 σ 2 ν 0 (∆V/∆ν 0 ) 2 S 2 l 1 τ (5)
with σ 2 P the PSD of laser power fluctuations, in W 2 /Hz, at f = f M (measured without the cell for a laser power P o impacting the photodiode) and ∆V/∆P is the fluctuation of the clock resonance signal induced by a laser power variation in V/W. We note also σ 2 ν 0 and ∆V/∆ν 0 the measured signal sensitivity to the laser frequency in V/Hz. Following the spectroscopy tests described above, we performed a detection noise analysis on both setups in order to evaluate the contribution of main noise sources on the laser beatnote short-term frequency stability. Corresponding noise spectra are shown, for both setups, in Fig. 6. Using equations reported above and resonance spectra shown in Fig. 2, Table 1 summarizes the key parameters of each laser setup and the respective contributions to the short-term noise budget. We calculate that the laser beatnote short-term stability should be limited at the level of about 1.6 × 10 -12 at 1 s by the DFB setup. In the DFB setup, the main noise contribution to the total budget is the laser phase noise through the intermodulation effect. The expected short-term stability at 1 s of the ECDL setup is estimated at the level of 7.8 × 10 -13 at 1 s and is currently mainly limited by the intermodulation effect and the FM-AM noise conversion process. We note that the contribution of the total detection noise of the ECDL setup has the potential to be reduced by about 10 dB by operating the ECDL setup with f m = 500 kHz instead of 62 kHz ( f m currently limited by the ECDL head controller electronics). In this case, the short-term instability of the ECDL setup should be reduced to a few × 10 -13 at 1 s. For confirmation of the results presented in Table 1, we have performed absolute phase noise measurements of the laser beatnote, shown in Fig. 7, in two different situations. In the first one, only the DFB laser is locked while the ECDL is free-running. In this case, the phase noise spectrum exhibits a clear f -3 slope, signature of a flicker frequency noise. Following the well-known power law such that S ϕ ( f ) = ∑ 0 i=-4 b i f i [START_REF] Rubiola | Phase noise and frequency stability in oscillators[END_REF], we extract b -3 = 93 dBrad 2 /Hz, yielding an expected Allan deviation σ y (1s) = 2 ln 2 b -3 ν 2 0 = 1.6 × 10 -10 . When both lasers are locked, the slope of the phase noise spectrum turns to f -2 , signature of the expected white frequency noise, for offset frequencies lower than about 1.5 kHz. In this case, we extract b -2 = 53 dBrad 2 /Hz, yielding an expected Allan deviation σ y (1s) = b -2 ν 2 0 = 9.4 × 10 -13 . This value is in agreement with the expected 1.6 × 10 -12 stability limit reported in Table 1 from the detection noise budget.

C. Short-term frequency stability

We performed a measurement of the laser beatnote short-term fractional frequency stability. In this test, the laser power is about 450 µW at the input of both cells and cell temperatures are fixed at 62 and 63 • C, for the ECDL and DFB, respectively. Results are shown on Fig. 8. The Allan deviation is 1.1 × 10 -12 τ -1/2 for integration times up to 100 s. This result is in good agreement with the short-term stability expected from the detection noise budget (1.6 × 10 -12 ) and absolute phase noise performances (9.8 × 10 -13 ), reported in section B. These stability results are competitive with those obtained in [START_REF] Newman | Architecture for the photonic integration of an optical atomic clock[END_REF][START_REF] Maurice | Miniaturized optical frequency reference for next-generation portable optical clocks[END_REF] (DFB laser-based Rb microcell optical frequency reference) for integration times up to 100 s and are encouraging for further exploration of this approach. They also compare favorably with recent results reported with DFSDS in a 10 cm long Rb vapor cell [START_REF] Zhao | Laser frequency stabilization via bichromatic Doppler-free spectroscopy of an 87 Rb D1 line[END_REF]. As suggested by our detection noise budget, the individual stability of the DFSDS-based microcell ECDL should be at the level of 8 × 10 -13 at 1 s in current conditions and should even be further improved by increasing the laser modulation frequency and servo bandwidth to a few hundreds of kHz. These previsions will be experimentally tested in a near future by comparing the present microcell-stabilized ECDL to an ultra-stable 895 nm signal generated from a lab-prototype cavity-stabilized laser [START_REF] Didier | Ultra-low phase noise all-optical microwave generation setup based on commercial devices[END_REF], with the help of an optical frequency comb. For integration times higher than 100 s, we currently observe a degradation of the clock stability, yielding the level of 1.9 × 10 -13 at 10 3 s and 6.3 × 10 -13 at 4000 s. We note that this result at 4 × 10 3 s is more than 10 times better than the one reported in our previous study [START_REF] Brazhnikov | Dual-frequency sub-Doppler spectroscopy: Extended theoretical model and microcell-based experiments[END_REF]. This improvement is probably due to an improved global thermal control of the experiment. The blue curve in Fig. 8 shows the deviation with application of a linear drift removal on experimental frequency data. In this case, the Allan deviation is slightly improved for τ > 250 s, yielding 2 × 10 -13 at 4 × 10 3 s. Studies are in progress to evaluate the contribution of main frequency shifts and improve the lasers' mid-term stabilities.

CONCLUSIONS

In conclusion, we have characterized the short-term stability budget of two lasers (one DFB laser and one ECDL) frequencystabilized with Cs microfabricated cells using the dual-frequency sub-Doppler spectroscopy (DFSDS) technique. The laser beatnote short-term frequency stability was measured at the level of 1.1 × 10 -12 at 1 s, in good agreement with the detection noise budget and absolute phase noise measurements. The short-term stability of the laser beatnote is limited by the intermodulation effect contribution induced by the frequency noise of the DFB laser. We believe that demonstrating a frequency stability at 1 s in the 10 -13 range, using a low noise ECDL locked to a microfabricated cell with a high servo bandwidth (a few 100 kHz), is a reasonable objective with the DFSDS approach. Further studies will be performed in a near future in this direction, combined with efforts to improve the laser mid-term stability, currently at the level of 6.3 × 10 -13 at 4 × 10 3 s. In addition, the enhanced compacity of the DFSDS-based laser system might be envisioned by using a directly-modulated laser [START_REF] Yun | High-performance coherent population trapping atomic clock with direct-modulation distributed Bragg reflector laser[END_REF] (to suppress the EOM), by replacing the AOM with a liquid crystal device or by benefiting from integrated photonics systems [START_REF] Wang | Nanophotonic lithium niobate electro-optic modulators[END_REF][START_REF] Sarabalis | Acousto-optic modulation in lithium niobate on sapphire[END_REF].

Fig. 1 .

 1 Fig. 1. (a) Schematic of the experimental setup made of two microcell-stabilized lasers using dual-frequency sub-Doppler spectroscopy (DFSDS). Yellow lines indicate fibered sections. See the text for description. (b) Photograph of a Cs microfabricated cell. (c) Photograph of the microcell physics package.

Figure 2 Fig. 2 .Fig. 3 .

 223 Figure2shows spectra of sub-Doppler resonances, detected in the bottom of absorption profiles, for both the DFB and ECDL setups, used in clock configuration. The total laser power at the input of the cell is about 450 µW. Due to a more favorable signal, our study was performed by locking the laser frequency to the Cs atom F = 4 excited state. In this test, the signal height S of

Fig. 4 .Fig. 5 .

 45 Fig.4. Signal height, FWHM and signal/FWHM ratio of the sub-Doppler resonance versus the cell temperature incident in the MEMS cell, for both laser setups. For this measurement, the laser power is 600 µW for both the DFB and ECDL setups.

Fig. 7 .

 7 Fig. 7. Absolute phase noise of the laser beatnote in different configurations: free-running ECDL vs locked DFB, or both locked.

Fig. 8 .

 8 Fig. 8. Overlapping Allan deviation of the laser beatnote, with or without drift removed.

  -11 for time scales up to 10 4 s. SDS was also explored in hybrid setups, in combination

	LO 4.596 GHz	Vbias	110 MHz	λ/2	λ/4
		EOM	AOM	Cs Cell 63°C	Mirror
		EOM	AOM	Cs Cell 62°C
		Vbias	80 MHz

Table 1 .

 1 Short-term noise and stability budget, for both DFB and ECDL setups. × 10 -12 5.75 × 10 -13

	Properties	DFB	ECDL
	S (mV)	126	139
	FWHM (MHz)	14.7	14.2
	S l (V/Hz)	8.6 × 10 -9	9.8 × 10 -9
	C	0.65	0.55
	Absorption (%) 63.1	58.6
	P i (µW)	450	450
	P o (µW)	166	186.5
	f M (kHz)	500	61.95
	Noise source	DFB	ECDL
	σ sn (1 s)	3.5 × 10 -15 3.8 × 10 -15
	σ pd (1 s)	3.9 × 10 -14 6 × 10 -14
	σ AM-AM (1 s)	3.8 × 10 -14 3.0 × 10 -13
	σ FM-AM (1 s)	4.4 × 10 -13 4.3 × 10 -13
	σ int (1 s)	1.5

σ y (1 s) 1.6 × 10 -12 7.8 × 10 -13

ACKNOWLEDGMENT

The authors thank Philippe Abbé and Gilles Martin (FEMTO-ST) for help with the development of the microcell physics package and electronics. The authors acknowledge F.-X. Esnault and J. Delporte (CNES) for their expertise and sharp appreciation of this study.

FUNDING

This work was supported in part by Centre National des Etudes Spatiales (CNES) through the OSCAR project (grant 200837/00), in part by Région de Franche-Comté with NOUGECELLL project (grant 20174-06245), and in part by Agence Nationale de la Recherche (ANR) in the frame of the LabeX FIRST-TF (Grant ANR 10-LABX-0048), EquipX Oscillator-IMP (Grant ANR 11-EQPX-0033) projects. The authors also thank the microfabrication platform MIMENTO and the Graduate School EIPHI from Université Bourgogne-Franche Comté. D. Brazhnikov was supported by Russian Science Foundation (grant 17-72-20089).

DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

DISCLOSURES

The authors declare no conflicts of interest.


A M Mikhailov

D V Brazhnikov

R Boudot

Analytical theory of dual-frequency sub-Doppler spectroscopy of alkali-metal atoms

Introduction:

Math model: Coherent population trapping (CPT):

Results:

In works [1,2] a new approach to the creation of miniature optical quantum frequency standards was proposed and investigated. To obtain a reference resonance in these works, colliding bichromatic laser beams with orthogonal linear polarizations are used. It was shown in [2] that, in such a configuration, it is possible to observe sub-Doppler resonances with a high contrast. Numerical simulations of these resonances were carried out in [3,1]. In this work we present some analytical results that provide a deeper understanding of the physics of the observed effects. Here, we use a simplified spectroscopic model based on a three-level Λ-scheme. The scheme is shown in figure 1. Λscheme interacts with a dual-frequency laser field composed of two counter-propagating, linearly polarized plane waves. Expression for the light field with two frequency components 𝜔 1 and 𝜔 2 :

𝐸 𝑧, 𝑡 = 𝐸 1 𝑒 -𝑖 𝜔 1 𝑡-𝑘 1 𝑧 + 𝑒 -𝑖 𝜔 2 𝑡-𝑘 2 𝑧 + 𝐸 2 𝑒 -𝑖 𝜔 1 𝑡+𝑘 1 𝑧+𝜑 1 +𝜑 + 𝑒 -𝑖 𝜔 2 𝑡+𝑘 2 𝑧-𝜑+𝜑 2 + C. C. The angle φ is the mutual angle between linear polarizations of counterpropagating waves. The angles φ 1, 2 is the spatial phase shifts.

The field E 1 pumps atoms with velocity -v into the noncoupled (dark) state |NC1⟩ while the field E 2 pumps those of velocity +v into the non-coupled state |NC2⟩ :

⟩ |𝑁𝐶 1 = 1 2 ⟩ |1 -𝑒 𝑖𝜔 12 𝑡+ 𝑖𝑘 12 𝑧 ⟩ |2 , ⟩ |𝑁𝐶 2 = 1 2 ⟩ |1 -𝑒 𝑖𝜔 12 𝑡-𝑖 𝑘 12 𝑧+𝜑 12 -2𝜑 ⟩ |2 .
Now, we consider the situation where the fields E 1 and E 2 start to interact with the same atoms. There are two main cases in this configuration:

(1) |NC 1 ⟩ and |NC 2 ⟩ interfere destructively, i.e. ⟨NC 1 |NC 2 ⟩ = 0;

(2) the 'competition' between |NC 1 ⟩ and |NC 2 ⟩ does not destroy the CPT effect, ⟨NC 1 |NC 2 ⟩ = 1. In the first case, atoms start to absorb intensively and scatter energy from the light field since they are no longer in the dark state.

Equation for one-atom density matrix in the Wigner representation:

𝑑 ො 𝜌 𝑑𝑡 = - 𝑖 ℏ 𝐻 0 + 𝑉, ො 𝜌 + 𝑅 ො 𝜌
The Hamiltonian 𝐻 0 of a free atom in the basis of eigenfunctions:

𝐻 0 = 𝑛=1 3 ℇ 𝑛 | ⟩ 𝑛 ⟨ | 𝑛 , 𝑉 = -𝐸 መ Ԧ 𝑑
The system of equations for the density matrix in the case of a linear approximation in the test field. : Population of the excited level in the approach of a standing wave:

𝜌
𝜌 33 = 2𝛾 𝑒𝑔 𝜏 𝑆 1 -+ 𝑆 2 + 1 + 2𝛾𝜏 + 2𝛾 𝑒𝑔 𝜏 2 1 + 2𝛾𝜏 2 𝑆 1 -2 + 𝑆 2 + 2 4 + 2𝛾 𝜏 2 -𝛽 + 𝑆 1 -𝑆 2 + 6 + 4𝛾 𝜏 1 -𝛽 + 2𝑅e -2𝛾 𝑒𝑔 𝜏 2 1 + 2𝛾𝜏 𝑆 1 -𝑆 2 + 𝑒 -𝑖 𝜃+2𝑘 12 𝑧
𝑆 𝑛 (±) = 𝑅 𝑛 2 /(𝛾 eg 2 + 𝛿 ± 𝑘𝑣 2 )saturation parameters, 𝐿 (±) = 1/(𝛾 eg + 𝑖 𝛿 ± 𝑘𝑣 )complex Lorentzian, 𝑅 = 𝐸𝑑/ℏ -Rabi frequencies, 𝛿one photon detuning, 𝛾 eg = 𝛾 + 𝛤 -relaxation rate of optical coherences with 𝛤 the time-of-flight relaxation rate attributed to the finite time of atom-field coherent interaction, 𝜏 = 𝛤 -1 , 𝜃 = 𝜑 1 -𝜑 2 -2𝜑.

Linear approximation in the test field: In this case, we consider the dependence of the absorption coefficient on the frequency detuning. The expression for the absorption coefficient will take the form: Figure 3 shows the graphs for each contribution and the dependence on the total 𝜃 phase. It can be seen from the presented ratio of the quantities that the CPT effect prevails over the effect from the saturated absorption resonance. If the cos(𝜃 + 2𝑘 12 𝑧) = -1 contributions from CPT and SAR have opposite values: SAR always leads to a dip. Since the value of the CPT contribution is larger, in this case a contrast absorption peak is formed in the center of the resonance curve, rather than a dip.

  1.Brazhnikov D. et al. Dual-frequency sub-Doppler spectroscopy: Extended theoretical model and microcell-based experiments //Physical Review A. -2019. -Т. 99. -№. 6. -С. 062508. 2.Brazhnikov D. V. et al. Two-frequency sub-Doppler spectroscopy of the caesium D1 line in various configurations of counterpropagating laser beams //Quantum Electronics. -2020. -Т. 50. -№. 11. -С. 1015. 3.Hafiz M. A. et al. High-contrast sub-Doppler absorption spikes in a hot atomic vapor cell exposed to a dual-frequency laser field //New Journal of Physics. -2017. -Т. 19. -№. 7. -С. 073028.

Fig. 1 Λ

 1 Fig. 1 Λscheme of atomic energy levels. Solid vertical arrows denote transitions induced by the light-wave components with vectors k 1, 2 , , while dashed vertical arrows are attributed to the waves with -k 1, 2 . Wavy black arrows denote spontaneous decay processes. 𝛾is relaxation rate, 𝛽is the openness coefficient.

+

  Averaging over the velocities of the thermal motion of atoms, we obtain the curve shown in Fig.2. The absorption coefficient can be represented as the sum of the Doppler and sub-Doppler parts: 𝛼 = 𝛼 Doppler + 𝛼 sub-Doppler .

Fig. 2

 2 Fig. 2 Dependence of the absorption coefficient on the frequency detuning δ, R=0.5𝛾, 𝛽 = 1𝛾, Г = 10 -2 𝛾, 𝑘𝑣 0 = 100𝛾

Fig. 4

 4 Fig. 4 Dependence of the population of the excited level on the frequency detuning δ. R = 0.05 𝛾, 𝛽 = 1𝛾, 𝛤 = 10 -2 𝛾, 𝑥 0 = 100𝛾


Bekara

M.D Nathalie Pham Dang

Isabelle Barthélémy

Farid Bekara

From rigid bone plate fixation to stable dynamic osteosynthesis in mandibular and craniomaxillo-facial surgery: historical evolution of concepts and technical developments

Keywords: stable dynamic osteosynthesis, evolution of concepts mandible fractures, biomechanics, miniplate, osteosynthesis, mandibular reconstruction fracture healing, mandible, rigid fixation, biomechanics, static osteosynthesis, dynamic osteosynthesis

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

Introduction

During the second half of the 20 th century major technological advances were made in France in the treatment of maxillofacial fractures. The Hamburg surgeon Carl Hansmann was the first to experiment the plate and screw system, in 1886. He was also the first advocate of rigid plate fixation in maxillofacial surgery, which until then had only been used for fractures of the extremities. Simultaneously, the Scottish William Arbuthnot Lane (1856Lane ( -1943)), in a publication of 1894 [START_REF] Verbeek | Osteosynthesis and the Lane technic[END_REF], and the Belgian Albin Lambotte (1866Lambotte ( -1955) ) [START_REF] Lambotte | The Classic. Contribution to conservative surgery of the injured hand[END_REF] pioneered internal fixation of displaced fractures. At the same time, Joseph Lister (1827Lister ( -1912)), the British surgeon and medical scientist, laid the foundations of antiseptic medicine [START_REF] Lister | On the Antiseptic Principle in the Practice of Surgery[END_REF]. However, because of the high rate of complications such as infection and facial nerve lesions osteosynthesis of mandible fractures was for many years put on hold. Almost a century later new concepts and progress in technical developments enabled oral and maxillofacial surgeons to advance from rigid bone plate fixation to stable dynamic osteosynthesis.

The aim of this presentation is to clarify which terms should be used for each of the different techniques of fracture treatment and to outline the principles which first led to the fixation techniques for mandible trauma.

Terminology

When discussing osteosynthesis it is of paramount importance to use uniform terminology.

For logical reasons, the name of a surgery technique should depend on its nature and results and not on the name, form or features of the material used. According to this principle, osteosynthesis can be rigid or dynamic and the result can be described as stable, unstable or solid. The term used should reflect the reality of the situation. This is why we propose the following definitions.

A fixation is said to be "unstable" when it needs to be completed by an intermaxillary fixation (IMF), which historically was the case of osteosynthesis with steel wire. This approach had the drawback of periosteal stripping of both sides of the mandible [START_REF] Lambotte | The Classic. Contribution to conservative surgery of the injured hand[END_REF] with impairment of the main blood supply to the bone [START_REF] Bradley | Fractures of the upper end of the femur. 2. Treatment[END_REF]. An unstable fixation is a failure.

In contrast, osteosynthesis is deemed to be "stable" when the broken limb can, before the fracture heals, be moved freely without application of force. For the mandible this means that the fixation does not require any associated IMF and that passive movements are possible and do not prevent the healing process. The fixation is stable at rest and during effortless movements. Liquid food is authorized [START_REF] Champy | Mandibular osteosynthesis by miniature screwed plates via a buccal approach[END_REF]. Preferably, only very light masticatory pressure should be applied.

The term "solid" refers to osteosynthesis that allows full operation of the mandible, including mastication, whether or not bone healing is complete. Most of the time, osteosynthesis is solid only when bone healing is complete.

The term "functionally stable" does not make a clear distinction between a stable and a solid fixation: both are functional but only one is solid.

The term "rigid" indicates that the plate absorbs any force applied on the operated mandible.

The plate can withstand all efforts exerted by the masticatory muscles without suffering any deformation. This term is sometimes used not only to designate the plate but also osteosynthesis itself or, in other words, the result of the process. Depending on the force applied a plate is qualified as either rigid, elastic or flexible. In contrast, synthesis can only be stable, solid or unstable. Mechanically speaking, the term "semi-rigid" does not make any sense.

Evolution of the concepts from orthopedic surgery to treatment of facial fractures

At the end of the 19 th century, the possibilities offered by surgical treatment of fractures drew the attention of oral and maxillofacial surgeons [START_REF] Vrebos | From circumferential wiring to miniaturized plates and screw. The history of osteosynthesis of the mandible with special reference to Dr A Lambotte[END_REF]. Lambotte coined the term osteosynthesis and established the ground rules for the technique: maintaining the bone's blood circulation, limiting dissection, and preserving the periosteal covering [START_REF] Lambotte | The Classic. Contribution to conservative surgery of the injured hand[END_REF]. He used a trapezoidal metallic resorbable plate made of nickel silver, a copper alloy with nickel and often zinc. His work had a considerable impact on orthopedic surgery but not on craniofacial surgery. The risk of lesions of the facial nerve during the cervical incision, the difficulty in restoring the dental occlusion owing to the rigidity of the plates, the risk of lesions of the alveolar nerve, the damage inflicted upon the tooth roots by the bicortical fixation of the screws and the high infection rate before the discovery of antibiotics impeded progress. Bradley et al. [START_REF] Bradley | Latent osteomyelitis; report of 7 cases[END_REF] stated that in the tooth portion, a fracture is a compound fracture directly communicating with the mouth cavity. They emphasized the importance of adapting and fully immobilizing the fractured surfaces (a mandatory requirement for recovery of the dental occlusion) and recommended that all teeth should be removed from the line of fracture to avoid chronic suppurative osteomyelitis. Since then, the treatment of fractures, malformations, cancers and bone fragment loss due to cancer or infection has become safer as control of the biological and mechanical factors that ensure better fixation and biologic healing has improved.

From clinical observation to empirical therapeutic applications

3.1.From bicortical to monocortical osteosynthesis

The first theories stated that the injured limb bone should be completely immobilized by rigid plates. The technique was first experimented in 1895 when Lane [START_REF] Lane | The operative treatment of fractures[END_REF] developed a metal plate to be used for internal fixation. Many authors, such as Aubry and Ginestet, advocated this technique in the treatment of fractures of the mandible [START_REF] Palfer-Solier | Chirurgie stomatologique et maxillo-faciale: Atlas de technique opératoire[END_REF][START_REF] Ginestet | Indications and techniques for the vertical osteotomy of the perpendicular rami of the mandible[END_REF][START_REF] Beal | Mandibular osteosynthesis[END_REF]. The approach is based on the idea that physiological bone healing is only possible when rigid fixation of the fracture is performed. It is accepted that bone healing is characterized by the absence of periosteal callus formation, called soudure per primam (primary bone healing). According to Müller and Perren, the appearance of any periosteal callus after plate fixation can be an indicator of an unknown degree of instability and infection [START_REF] Müller | Callus and primary bone healing[END_REF].

In line with these precepts, Danis developed numerous techniques of osteosynthesis based principally on interfragmentary compression using screws and a device that he called "coapteur", which was basically a plate designed to produce axial compression between two main bone fragments [START_REF] Danis | Théorie et pratique de l'ostéosynthèse[END_REF]. His model was extrapolated by Luhr [START_REF] Luhr | On the stable osteosynthesis in mandibular fractures[END_REF] and Spiessl [START_REF] Roth | Treatment of extensive bone defects in the jaws with hydroxyapatite granules. Schweiz Monatsschrift Zahnmed[END_REF] to the management of mandibular fractures: stability was enhanced by increasing the friction forces between fracture surfaces and using specific rigid plates with bicortical screws specially designed to provide inter-fragmentary compression. This method consists in stable internal fixation of fractures by an eccentric dynamic compression plate (EDCP).

In 1967, Franchebois and Souyris started using modified Müller plates, a design that allows inter-fragmentary compression in the treatment of fractures of the mandible by tightening a tensor temporarily anchored to the bone and the plate [START_REF] Franchebois | 6 years of experience in osteosynthesis of anterior fractures of the mandible by infra-oral approach[END_REF][START_REF] Souyris | Mandibular osteosynthesis with screwed plates without blocking: original method and results after 5 years[END_REF]. In this approach, the plates are fixed to the inferior border of the mandible. As a result of stable internal fixation, intermaxillary fixation was no longer necessary. The technique allowed free mobility of the mandible and a soft or semi-solid diet. Initially, it was restricted to subjects contraindicated to IMF such as epileptics and edentulous patients and in cases of pseudoarthrosis and plurifocal fractures.

Thus, although intermaxillary fixation was no longer a requirement, new challenges emerged: the difficulty to concomitantly control dental occlusion and fracture reduction, and avoid disjunction of the alveolar rim at the fracture angles.

To counter the problem, Michelet along with Franchebois and Souyris developed the intraoral approach [START_REF] Franchebois | 6 years of experience in osteosynthesis of anterior fractures of the mandible by infra-oral approach[END_REF][START_REF] Michelet | Osteosynthesis with miniaturized screwed plates in maxillo-facial surgery[END_REF][START_REF] Michelet | Surgical treatments of fractures of the corpus mandibulae without blockage, with diminutive screwed plates inserted via the endobuccal route[END_REF]. The method avoids skin scars, facial nerve damage, restores occlusion, allows traumatized teeth to be treated and direct observation of the alveolar bone.

As a result of the endo buccal incision, the plates could not be placed at the inferior border of the mandible, and because of the position of the tooth roots bicortical screws could no longer be used. Souyris and Michelet suggested placing the plates on the upper border of the mandible using monocortical screws to avoid alveolar rim disjunction and systematic tooth lesions while respecting the principle of rigid fixation [START_REF] Souyris | Mandibular osteosynthesis with screwed plates without blocking: original method and results after 5 years[END_REF][START_REF] Michelet | Osteosynthesis with miniaturized screwed plates in maxillo-facial surgery[END_REF].

From the principle of rigid compression fixation to stable dynamic osteosynthesis

As in skin surgery, compression of a biological tissue damages its blood supply. Sustained compression on the fracture surfaces can suppress or at least reduce the blood supply. For these reasons, continuous compression seems illogical and inappropriate. Enhancing or disturbing bone healing in the area adjacent to the fracture surfaces depends on how much compression is exerted on the fracture surfaces, under the plate and around the screws. The so-called "dynamic compression" of Luhr and Spiessl was in fact a single mechanical compression while the term "dynamic" implied several repeated micro-movements caused by muscular activity. Although Luhr's plate was called a dynamic compression plate (DCP) only one-time static compression could be obtained. This stage in the development was still empirical and based on clinical observations. In 1870, Wolff is credited with originating the trajectory relating bone structure to the mechanical forces imposed upon it: where stresses of pressure and tension occur in bone, formation of bone takes place [START_REF] Wolff | The classic: on the inner architecture of bones and its importance for bone growth[END_REF]. A hundred years later, McKibbins (1978) advanced the principle of "secondary fracture healing" [START_REF] Mckibbin | The biology of fracture healing in long bones[END_REF]. This process is possible when complete rigidity is not achieved, which results in greater motion at the fracture site and a degree of intermittent displacement between the bone ends. Healing progresses via the three-stage process of inflammation, callus formation and remodeling. The goal of this complex process is a stepwise increase in the mechanical stability of the fracture site that is achieved by progressively replacing fragile provisional tissues with more stable ones, eventually reaching a point that allows vascular ingrowth and mineralization processes to occur. Finally, the procedure of stable dynamic osteosynthesis was adopted and the technique of rigid fixation was abandoned, thereby introducing a new rationale in the treatment of fractures.

3.3.The concept influences the design of the plate and not vice versa

Many publications have focused on the study of plates, their form, dimension, nature and tolerance [START_REF] Vrebos | From circumferential wiring to miniaturized plates and screw. The history of osteosynthesis of the mandible with special reference to Dr A Lambotte[END_REF]. Defining the concepts underlining the treatment methods is more important than stating the nature or dimensions of the material used. Maxillofacial plates and screws are smaller than the material used for limb surgery. Thus, once the concept of stable-dynamic osteosynthesis was adopted, the first publications used the term miniplate osteosynthesis as opposed to material used for limb fractures. The use of the term miniplate quickly became widespread and was soon universally accepted [START_REF] Michelet | Osteosynthesis with miniaturized screwed plates in maxillo-facial surgery[END_REF]. To date osteosynthesis of mandible fractures has been qualified by the material used and not by the technique. Adaptive osteosynthesis, which means restoring the morphology of the fracture by precisely adapting its surfaces would have been a more appropriate term.

The term "miniplate osteosynthesis" led to problems when Michelet began to work with resorbing plates [START_REF] Sedel | Biodegradable implants in orthopedic surgery[END_REF][START_REF] Heidemann | In vivo study of degradation of poly-(D,L-) lactide and poly-(L-lactide-co-glycolide) osteosynthesis material[END_REF]. The procedure still corresponded to adaptive osteosynthesis but the term "miniplates" was a misnomer because of the dimensions of the resorbable plates [START_REF] Michelet | Surgical treatments of fractures of the corpus mandibulae without blockage, with diminutive screwed plates inserted via the endobuccal route[END_REF].

From experimental studies to stable dynamic osteosynthesis.

During a period that could be called the empirical stage the methods used were a direct application of contemporary knowledge of general orthopedic surgery to the mandible.

There then followed a period in which the compression principle was extrapolated to mandibular surgery and the techniques used followed or preceded knowledge of the bone healing process [START_REF] Danis | Théorie et pratique de l'ostéosynthèse[END_REF][START_REF] Uhthoff | Internal plate fixation of fractures: short history and recent developments[END_REF]. New developments in this third stage, the experimental period, arose mainly from biomechanical studies undertaken in Strasbourg from 1972 to 1974 by the Groupe d' Etude en Biomécanique Ostéo-Articulaire de Strasbourg (GEBOAS), who not only created a new concept but also described the features of the plates and screws to be used for the method [START_REF] Lambotte | The Classic. Contribution to conservative surgery of the injured hand[END_REF][START_REF] Lane | The operative treatment of fractures[END_REF][START_REF] Danis | Théorie et pratique de l'ostéosynthèse[END_REF][START_REF] Uhthoff | Internal plate fixation of fractures: short history and recent developments[END_REF][START_REF] Sherman | Vanadium steel bone plates and screws[END_REF]. The GEBOAS performed static and dynamic experiments to establish objective principles in mandibular osteosynthesis.

4.1.The cortical bone

In the late 1970s, studies described the external surface of the mandible, calculated the distance between each dental root tip and external surface of the outer cortical layer of the mandible and to the inferior border of the mandible and clarified the position of dental root tips in relation to the occlusal line and the inferior alveolar nerve. The observations showed that the cortical bone of the mandible had an average thickness of 3.3 mm and that the basilar area is thicker and the apical alveolar area considerably thinner [START_REF] Champy | Mandibular osteosynthesis by miniature screwed plates via a buccal approach[END_REF] .

Clenching forces

The GEBOAS measured the breaking load of an isolated mandible with an Adamel Lhomargy machine and were then able to determine the resistive forces of the plates and the bite force.

The value obtained helps to determine the mechanical resistance of the plates. Torsional moment, or torque, was also analyzed in the curved symphyseal region [START_REF] Champy | Mandibular osteosynthesis by miniature screwed plates via a buccal approach[END_REF] (Fig. 1).

4.3.The localization of plates and screws on a fractured straight beam

Once the clenching forces had been measured, it was necessary to determine the distribution of the strains developed inside the mandible. When pressure is applied on the free end of a straight unilaterally fixed beam, traction strains are recorded on the upper border and compression strains are visible on the lower border. However, at the point of force application, forces are shifted. When a load is placed on the extremity of the fractured straight beam, if the fracture is fixed in the lower border of the beam, compression and traction strains are recorded on the lower border and diastasis occurs at the upper end of the fracture line. If the plate is located along the upper border of the beam, traction strains are neutralized.

Compression strains remain concentrated at the lower border. No disjunction occurs (Fig. 2).

When a force is directly applied to the upper extremity of the fracture line, compression strains are observed at the point of application of the force and traction strains at the lower border, the opposite of what is observed when the force is applied at a distance [START_REF] Champy | Mandibular osteosynthesis by miniature screwed plates via a buccal approach[END_REF][START_REF] Champy | Mandibular osteosynthesis and compression[END_REF]. This neutralizes traction stresses and reestablishes the normal run of concomitant compression stresses. The later works of Farmand reported that the direction of torsion and traction forces was reversed at the impact point of a clenching force. Subsequently, studies of a curve beam subjected to a load showed that torsion strains could be neutralized by two plates [START_REF] Farmand | Experiences with the 3-D miniplate osteosynthesis in mandibular fractures[END_REF]. Thus, a fracture of the mandible behind the canine can be fixed with a single plate whereas a fracture of the symphysis requires two plates to neutralize the torsion forces exerted in the curved area (Fig. 3) [START_REF] Ellis | An analysis of 2,067 cases of zygomatico-orbital fracture[END_REF].

Periosteal bone healing

Other investigations were performed to determine whether the upper position of the plates is transferable to the mandible, which is a curved model. A study carried out under the supervision of M. Grossman of the Laboratoire de spectroscopie et de photonique iconique de l'Institut de Physique (Pharok Piam) [START_REF] Champy | Mandibular osteosynthesis according to the Michelet technic. II. Presentation of new material[END_REF] used a laser beam to examine recently collected post-mortem fractured and synthesized mandibles. Two cameras were placed in a rigid position and centered on the fracture line of the mandible. One camera was lit by a laser flash light with the image obtained serving as a reference. The osteosynthetized mandible was then subjected either to traction strains or to torsion strains. Further photographs were taken and merged with the reference images to measure with greater accuracy the interference fringe caused by superimposition. These results showed micro-movements between the osteosynthetized fragments, which was contrary to the expected result of strict immobilization principles that had been recommended until then. They also demonstrated that in no case could the new mode of fixation be considered as rigid. These observations confirmed that mechanical stimulations spread from one fragment to another help the healing process by recreating conditions similar to bone physiology [START_REF] Sarmiento | Prefabricated functional braces for the treatment of fractures of the tibial diaphysis[END_REF][START_REF] Cornell | Newest factors in fracture healing[END_REF].

A case series involving two patients suffering from a mandibular fracture secured with a plate included direct examination of their stabilized fractures.

On the 10th day, surgical exposure of the fracture site showed that the periosteum had completely covered the fracture line, screws and plates. The periosteum had thickened and showed no signs of inflammation. Light and careful pressure on the dental crown, at the top of the fracture line, showed slight, fleeting diastasis of the inferior border of the mandible.

This direct observation of the surgical site demonstrated that bone healing was of a periosteal nature and not primary bone healing. It also confirmed that the different strain forces were reversed at the impact point of a masticatory force applied to the fracture line. Stable dynamic fixation of a mandibular fracture allows recovery of the physiological condition of the bone [START_REF] Champy | Mandibular osteosynthesis by miniature screwed plates via a buccal approach[END_REF].

4. 5 .

 5 The ideal plates and screws for stable dynamic osteosynthesis Once the location of the plates and screws has been established, resistance of the external cortical bones to the pull-out forces applied on the screws can be assessed. The choice of osteosynthesis material should be compatible with the specific biological and mechanical imperatives of the mandible: tolerance, solidity and ductility. GEBOAS used a modified Young's modulus to achieve the proper rigidity and flexibility of the metal. The rigidity, elastic limits and breaking point of material under loading were then measured. Bendable stainless steel was the first suitable material to be used but was progressively replaced by titanium. The shape of the plate (thickness of 1.00 mm, length varying from 4 holes with or without a bar) and accommodation of the screw holes were influenced by the design of Shermann et al. The characteristics of the screws depend on the thickness and structure of the cortical bone. Different materials have been used to manufacture the plates and screws.Hansmann used gold and Lamotte nickel silver (resorbable). Later came steel, stainless steel, titanium and finally resorbable Phusiline plates and resorb plates, developed by Champy and Gerlach, respectively[START_REF] Michelet | Surgical treatments of fractures of the corpus mandibulae without blockage, with diminutive screwed plates inserted via the endobuccal route[END_REF][START_REF] Heidemann | In vivo study of degradation of poly-(D,L-) lactide and poly-(L-lactide-co-glycolide) osteosynthesis material[END_REF][START_REF] Champy | Osteosynthesis using resorbable plates in maxillofacial surgery: hopes and disappointments[END_REF]. Michel et al. started using Phusiline, which is perfectly tolerated, in 1971 but later abandoned the technique[START_REF] Michelet | Osteosynthesis with miniaturized screwed plates in maxillo-facial surgery[END_REF][START_REF] Altemir | Sub-mental vs. retromolar intubation[END_REF].CONCLUSIONFor a long time the aims of surgical treatment of the fractures of the jaw using plates and screws were always the same: restore the occlusion and ensure bone healing. The historical evolution of the principles of treatment and the technical developments of the treatment methods occurred in three stages. The first stage, which should be called adaptive osteosynthesis, consisted in strict immobilization of the fracture with rigid plates and a bicortical fixation. The surgical approach was extra-oral with a high risk of damaging teeth roots, facial nerves and the inferior alveolar nerve. Restoring the occlusion was problematic.The second stage, rigid compression osteosynthesis, required the development of special rigid compressive plates derived from the coapteur of Danis. The method consisted in increasing stability by increasing compression and friction forces. Restoration of the occlusion was not easy and damage to the inferior alveolar nerve was frequent. Bone healing was of the primary healing type. The third stage, stable dynamic osteosynthesis, consisted in the restoration of the lines of force in the mandible and the transfer of micro-movements between the two fragments. The material used is adaptable, which results in easy protection of the nerves and teeth and restoration of the occlusion The biomechanical study performed by the Strasbourg bone and joint research group (GEBORS) showed that fixation of a bone structure had to be performed with a technique that neutralized the traction and rotation forces and restored the compression and friction forces.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Torsion moments in the symphysal region

Acknowledgment and gratitude to Emerit Professor Professor Maxime Champy for its essential assistance in this work. I hope that this work reflects a fair and accurate vision of your thought.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.