METHOD FOR SERVO CONTROL OF AN OPTICAL DEVICE WITH LASER AND RESONATOR FOR COMPENSATING AMPLITUDE MODULATION INTRODUCED BY A PHASE MODULATOR

Description

Technical field :

[0001] The invention relates to the field of controlling the laser optical frequency on a resonant optical cavity or vice versa. Previous technique:

[0002] In many applications (gyrometer, gravitational interferometry or metrology), it is necessary to have a laser source emitting radiation at a very stable optical frequency. In order to improve the stability of the optical frequency of a laser, it is known to control the laser on the resonance of a resonant optical cavity.

[0003] The most commonly used method to achieve such control is the so-called Pound Drever Hall method, named after its inventors (see RWP Drever, Laser Phase and Frequency Stabilization Using an Optical Resonator, Appl. Phys. B 31, 97-105, 1983). figure 1schematically represents a device for carrying out this method. The device comprises in particular a cavity C having a resonance around a central frequency fc and a laser L adapted to generate a laser beam LL. The optical frequency of the laser beam is then shifted by an acousto-optic modulator AOM (optional component) to a frequency f = oh / 2π included in said resonance. The laser beam is then modulated, by a PM phase modulator, by a phase at a modulation frequency f-mod = ω mod / 2π and a modulation amplitude M. The modulation frequency is applied by a local oscillator OL.

[0004] By a harmonic development, it can be shown that the phase-modulated beam injected into the cavity has an amplitude E inj such as : E inj = J 0 M e iωt + J 1 M e i ω + ω mod t − J 1 M e i ω − ω mod t , with JBessel functions, this equation being valid for low M by limiting the development to the first sidebands.

[0005] This equation allows us to observe that the PM phase modulator creates lateral components, "side-bands", in frequency separated from the initial frequency f by multiples of the modulation frequency, f mod , applied by the OL oscillator via PM. This modulation frequency is chosen, if possible, to be greater than the width of the cavity resonance so that the sidebands are not in resonance with the cavity. In the following, only the two sidebands are considered f ± f mod .

[0006] The laser beam is then injected into cavity C. The latter has a reflection transfer function noted F r which links the amplitude of the incident electric field and that of the reflected electric field. An optical circulator CO (typically based on a Faraday rotator placed between two polarization separator cubes) directs the laser beam reflected by the cavity onto a photodiode denoted PDr. The signal SI generated by the photodiode is then demodulated by a demodulation system Dem 0 comprising a mixer Mix 0 multiplying the signal SI by the modulation signal applied to PM with a demodulation phase. A low-pass filter LP 0 then makes it possible to keep only the DC component of the demodulated signal whose amplitude, Vε, called the error signal, is then proportional to the difference between the frequency f of the laser and the frequency fc cavity resonance. In fact, the amplitude of the error signal is: V ϵ = 2 G P c P s Im F r ω . F r ω + ω mod ¯ − F r ω ¯ . F r ω + ω mod , with P c and Psthe power of the fundamental and sideband components respectively and G the conversion gain between the received optical power and the voltage delivered by the photodiode

[0007] When the modulation frequency is sufficiently high and when a deviation δv between the frequency of the laser radiation and the central frequency of the resonance is sufficiently small, then it can be shown that: V ϵ = G 8 P c P s Δ f c δν , with Δf c , the resonance line width (for a high finesse Fabry-Pérot cavity).

[0008] When the frequency f of the laser and the resonance frequency of the cavity deviate a little, the two side-bands are unchanged (if they are indeed off resonance) while the phase and the amplitude of the beam at the frequency fevolve (since it is no longer at resonance). The coherence properties between the three spectral components of the laser beam then allow a measurement of these fluctuations which result in this linear variation of the demodulated signal which can thus be used as a frequency error signal, Vε being cancelled when the laser radiation is resonant with a mode of the cavity.

[0009] A UT processor is then configured to perform a servocontrol with this error signal V , via feedback electronics, according to classic servo-control methods, for example, without being restrictive, with PI or PID feedback electronics for Proportional Integral Derivative, an allusion to the three modes of action on the error signal of the feedback electronics. This type of feedback allowing the error signal to converge towards a zero value is well known in automation.

[0010] Concerning the choice of the modulation frequency to be applied to PM, if the cavity finesse is large, the width of the cavity resonances will be small compared to the free spectral interval and the modulation frequency can be chosen very large compared to the frequency width of the cavity resonance peaks. We will then be in the optimal situation, corresponding to the previous explanation, for this control. On the other hand, if the cavity finesse is not very large, the modulation frequency will be close to the frequency width of the cavity resonance peaks. The sidebands are then partially modified when the frequency f deviates from resonance and the control is less efficient.

[0011] The control loop feeds back to the laser, for example via the DC voltage supplying the laser (in the case of a semiconductor laser, for example) so as to maintain or conserve the frequency of the laser. fon a resonance frequency of the cavity. Alternatively, the control is carried out on the length of the cavity, the frequency of the laser remaining fixed. More precisely, the device comprises a piezoelectric translation stage (not shown in figure 1 ) on which one of the cavity mirrors is fixed for example, in order to control a length of the cavity or to extend or contract a part of the cavity if it is a fiber cavity. A final option consists of carrying out the control on the AOM via the excitation frequency of the AOM.

[0012] This technique works theoretically perfectly. However, in practice, it suffers from a major flaw. Indeed, most phase modulators are based on electro-optical crystals (for example those made of Lithium Niobate) and introduce an amplitude modulation associated with the phase modulation (residual amplitude modulation, RAM). Indeed, the medium used being birefringent, this modulation is due to the fact that the incident polarization of the laser radiation is not perfectly aligned with the used axis of the crystal. At the output of the modulator, the electric field has two components linked to the importance of the projection of the incident electric field on the two axes of the crystal.Furthermore, it is common that at the output of the phase modulator, a more or less perfect polarizer is included on the path (because a coupler or an isolator or even a circulator is inserted behind the phase modulator or because the polarization state must be well defined for needs related to the experiment and then a polarizer is introduced). After projection on the passing direction of the polarizer, the mixing of these two components induces an amplitude modulation associated with the phase modulation.

[0013] The problem is that both of these phase and amplitude modulations are at the same frequency f mod .The demodulation process explained above therefore does not allow them to be separated and the amplitude modulation results in a bias on the error signal. When this error signal is used by a servo, this bias results in the laser being locked next to the cavity resonance or absorption peak. In addition, this bias can vary because it is sensitive to temperature as it depends on the natural birefringence of the crystal.

[0014] In order to correct the RAM, the most classic method consists, at the output of the PM phase modulator, in taking, by means of an LSP splitter plate, a part of the phase-modulated LL laser beam and in detecting it by a photodiode PD ram so-called RAM control (see figure 1). By demodulating at the modulation frequency, it is then possible to generate a signal proportional to the induced bias. This signal is used to counteract on the crystal via the UT processor by applying, in addition to the modulation voltage, a DC or slowly variable voltage depending on the fluctuations of the bias. This DC voltage creates a birefringence which opposes the natural birefringence of the crystal. The bias is thus correctly canceled on this photodiode PD ram.

[0015] The major flaw of this approach is that this photodiode PD ram is not the photodiode PD r which allows the laser L to be controlled on the cavity C. However, in the case where the assembly includes other birefringent components (optical fibers, circulators, couplers, etc.) after the PM phase modulator, cancel the modulation amplitude on the photodiode PD ram does not guarantee its cancellation on the photodiode PD r.

[0016] There figure 2 illustrates this problem more precisely in a non-limiting example. In this figure, the device comprises a Pol polarizer at the input before the PM phase modulator which is followed by a 1x2 coupler coupled to two polarization-maintaining fibers. One fiber is redirected to PD ram (not shown in figure 2 ) and the other FMP fiber to the cavity C through the CO circulator including a polarizer. The CO polarizer allows for example to fix the state of the polarization of the incident field on the cavity in order to avoid introducing biases or instabilities due to the deformation of the resonances by the excitation of a bad polarization. The field leaving the phase modulator is written (ordinary axis along y and extraordinary axis along z): E PM → = E 0 e iωt e iϕ 3 cos β z → + e iϕ 2 sin β y → , with ϕ 2 , 3 = ϕ o , e + ϕ o , e dc + δ o , e sin ω mod t , the phase applied by the phase modulator ϕ o , e + ϕ o , e DC = 2 πl λ n o , e − πl λd r o , e n o , e 3 V DC , with no, nothe indices according to the ordinary and extraordinary axes respectively, and V DC the DC voltage applied to the phase modulator δ o , e = − πl λd r o , e n o , e 3 V RF , with l,d the dimensions of the phase modulator and V RF the alternating voltage applied to the phase modulator and f-mod = oh mod | 2π the modulation frequency.

[0017] We recall that for a Pound-Drever-Hall type servo, the optimal modulation depth M = δ e is approximately 1 rad and taking into account the values ​​of no = 2.2108, no = 2.1373, r o = 8.6pm / V, re = 30.8pm / V, we have δ o = δ e r o n o 3 r e n e 3 ≈ 0.3

[0018] Either β' the angle between the z axis of the phase modulator crystal and the z' axis of the FMP fiber so that: z → = cos β ′ z ′ → + sin β ′ y ′ → y → = − sin β ′ z ′ → + cos β ′ y ′ → E PM → = E 0 e iωt e iϕ 3 cos β cos β ′ z → ′ + sin β ′ y → ′ + e iϕ 2 sin β − sin β ′ z → ′ + cos β ′ y → ′ = E 0 e iωt e iϕ 3 cos β cos β ′ − sin β sin β ′ e iϕ 2 z → ′ + e iϕ 3 cos β sin β ′ + e iϕ 2 sin β cos β ′ y → ′

[0019] After crossing the FMP fiber, characterized by its index n slow along the z axis and n fast along the y axis: E FMP → = E 0 e iωt e iϕ 3 cos β cos β ′ − sin β sin β ′ e iϕ 2 e iϕ s z ′ → + e iϕ 3 cos β sin β ′ + e iϕ 2 sin β cos β ′ e iϕ f y ′ → , with f f,s the phase shift induced by the FMP fiber along the y and z axes respectively, then after the CO polarizer oriented to y' of the z' axis, the field Yes is written: E CO → = E 0 e iωt e i ϕ s + ϕ f 2 e iϕ 3 cos β cos β ′ − sin β sin β ′ e iϕ 2 e i ϕ s − ϕ f 2 cosγ ′ + e iϕ 3 cos β sin β ′ + e iϕ 2 sin β cos β ′ e − i ϕ s − ϕ f 2 sinγ ′ γ ′ → en posant Δ ϕ ′ = ϕ s − ϕ f 2 = E 0 e iωt e i ϕ s + ϕ f 2 e iϕ 3 cos β cos β ′ cosγ ′ e i Δ ϕ ′ 2 + sin β ′ sinγ ′ e − i Δ ϕ ′ 2 + e iϕ 2 sin β − sin β ′ cosγ ′ e i Δ ϕ ′ 2 + cos β ′ sinγ ′ e − i Δ ϕ ′ 2 γ ′ → E CO → = E 0 e iωt e i ϕ s + ϕ f 2 b ′ e iϕ 3 + a ′ e iϕ 2 γ ′ → = E 0 e iωt e i ϕ s + ϕ f 2 b ′ e i ϕ 3 + ϕ b ′ + a ′ e i ϕ 2 + ϕ a ′ γ ′ → by introducing b ′ = cos β cos β ′ cosγ ′ e i Δ ϕ ′ 2 + sin β ′ sinγ ′ e − i Δ ϕ ′ 2 = b ′ e iϕ b ′ a ′ = sin β − sin β ′ cosγ ′ e i Δ ϕ ′ 2 + cos β ′ sinγ ′ e − i Δ ϕ ′ 2 = a ′ e iϕ a ′

[0020] The intensity on the RAM control photodiode is then: I T = E CO . E CO ¯ = E 0 2 a ′ 2 + b ′ 2 + 2 b ′ a ′ cos ϕ 3 − ϕ 2 + ϕ b ′ − ϕ a ′

[0021] With ϕ 3 − ϕ 2 = 2 πl λ n 0 − n e − πl λd r o n o 3 − r e n e 3 V DC − πl λd r o n o 3 − r e n e 3 V RF . sin ω mod t

[0022] By asking: Δ ϕV + Δ ϕ DC = ϕ o − ϕ e + ϕ o DC − ϕ e DC = 2 πl λ n 0 − n e − πl λd r o n o 3 − r e n e 3 V DC

[0023] By decomposing into harmonics, then selecting the first harmonic and demodulating the signal generated by the Dem system to oh mod, with a demodulation phase chosen to be zero, we obtain a RAM error signal V RAM to cancel which is worth: V RAM = − 2 a ′ b ′ E 0 2 J 1 M sin Δ ϕ + Δ ϕ dc + ϕ b ′ − ϕ a ′ , with J 1 the Bessel function of order 1.

[0024] So the tension continues V DC to apply to the modified phase modulator Δφ dc< and must correct both the birefringence of the crystal and that of the fiber ( f b '- φ a' ) so that VRAM = 0 i.e. sin(Δ f + Δφ dc< + φ b' - φ a' ) = sin(Δ f RAM ) = (2k + 1) π / 2 . This is of course possible because the correction is modulo 2π but the problem is that the part of the beam that propagates in the other fiber from the coupler to the cavity will undergo other fluctuations. The correction on the RAM control photodiode therefore does not allow the correction on the other beam. This analysis can be carried out in other cases, for example if a polarizer is present between the phase modulator and the optical fiber. They lead to the same condition sin( Df + Δφ dc< + φ b' - φ a' ) = kπ with a definition of φ b' - φ a'which depends on the assembly. The conclusion is thus identical: cancellation on one arm of the assembly does not guarantee cancellation on another arm except possibly in free space with the absence of polarizing elements. Another limitation common to a free space or fiber assembly in which a photodiode is dedicated to RAM control and another to laser and cavity locking is the demodulation phase. In fact, to benefit as much as possible from RAM cancellation, the demodulation phase used to generate the error signal used to lock the laser must be the same as that used to cancel the RAM on the RAM control photodiode. If this is not the case, the demodulation process reveals a component linked to the RAM. In practice, it is not easy to guarantee this equality of phases.The demodulation phases, which are adjusted to the synchronous detections, in fact correct the delays due to the electronics (amplifier at the photodiode output, propagation in the cables, in the synchronous detection, etc.) and these phases can drift.

[0025] The invention aims to overcome some of the aforementioned problems of the prior art. More specifically, the invention relates to a method for performing the RAM correction on the same photodiode as that used to perform the servocontrol of the laser optical frequency on the resonance of the cavity. This method therefore makes it possible to overcome an error introduced by the birefringence of the components introduced after the phase modulator.

[0026] Furthermore, the article "Suppression of residual amplitude modulation of ADP electro-optical modulator in Pound-Drever-Hall laser frequency stabilization" (ISSN: 0030-3992, DOI: 10.1016 / J.OPTLASTEC.2020.106758) teaches a measurement and analysis of residual amplitude modulation, RAM, using a photodiode detecting the radiation reflected by a resonant cavity and differentiating two demodulation signals. Summary of the invention :

[0027] To this end, an object of the invention is a method of controlling an optical device comprising a cavity exhibiting a resonance around a central frequency. fc , a laser and a phase modulator, said method being adapted to control said cavity on said laser or vice versa and to compensate for an amplitude modulation introduced by said phase modulator, said method comprising the following steps: Generate, by said laser, laser radiation at an optical frequency f included in said resonance Vary a gap dv between said optical frequency of said laser radiation and said central frequency so that said optical frequency sweeps said resonance, said deviation being controlled by a parameter of an element of said device, and for each deviation dv io Modulate, at a modulation frequency f mod , a phase of said laser radiation, by a modulation phase φ mod , with the phase modulator, o Inject the phase-modulated radiation into said cavity, o Detect, by a photodiode, radiation reflected or transmitted by said cavity and generate an electrical signal representative of the intensity of said detected radiation, o Demodulate said electrical signal at said modulation frequency f-modby synchronously generating a first demodulated signal and a second demodulated signal representative of the demodulated electrical signal, respectively at a first demodulation phase f dem, 1 and a second modulation phase φ dem,2 φ dem,2 ≈ f dem, 1 k< , with k ∈ [0; 2π] different from said first phase and by filtering the first and second signals so as to retain only a continuous component of the first demodulated signal V 1 called error signal 1 and the second demodulated signal Vé 2, called error signal 2 C. Calculate a function 1 f 2 and a function 2 f 2 representing respectively an evolution of the error signal 1 and an evolution of the error signal 2 as a function of said deviation for a given value of said first demodulation phase, D. Repeat steps B) and C) by varying said first demodulation phase between each repetition until, for a value of the first demodulation phase called the decoupling phase f dem, 1 = φdec ,said function 1 or said function 2 has a plateau on a portion of values ​​of said deviation comprising 0, said function among 1 or 2 having the plateau being called PL function, the function not having the plateau being called SP function; E. In said decoupling phase, varying said deviation so as to observe an extremum, for a so-called zero deviation, said intensity of the light radiation detected by said photodiode and controlling said element on a value of said parameter making it possible to maintain this intensity extremum, F. In said decoupling phase, modulating the phase of said laser radiation, with the phase modulator (PM), by a so-called additional periodic signal at a so-called additional modulation frequency add in addition to that at the said modulation frequency f-mod; G. Minimize, for the SP function, an amplitude of a first harmonic of the additional signal, by varying said first demodulation phase, this minimum being reached for a first demodulation phase called additional φ dem,1 = φ dec2 , and, at said additional phase, control said element to a value of said parameter making it possible to maintain this intensity extremum H. At said decoupling phase f dem, 1 = φ dec and at said zero deviation, varying said first modulation phase until the PL function is cancelled, and maintaining the cancellation of said PL function by a control of said phase modulator.

[0028] According to a particular embodiment of the method of the invention, step H) consists of varying a voltage V dc,pm of said phase modulator up to a voltage value called RAM allowing the PL function to be cancelled and said phase modulator to be controlled by said RAM voltage.

[0029] According to another particular embodiment of the method of the invention, compatible with the previous embodiment, in step iii), said photodiode detects the radiation transmitted by said cavity and said method comprises a step D1) after step D) and before step E) consisting of varying said modulation frequency until a slope of said SP function is maximum on said portion of values, by repeating step D for each modulation frequency.

[0030] According to another particular embodiment of the method of the invention, compatible with the two previous modes, the additional frequency add is 10 times lower than the modulation frequency f mod .

[0031] Another object of the invention is an optical device comprising a cavity exhibiting a resonance around a central frequency fcand a laser adapted to generate laser radiation at an optical frequency f included in said resonance, said device comprising: an element suitable for varying a gap dv between said optical frequency of said laser radiation and said center frequency such that said optical frequency sweeps said resonance, said deviation being controlled by a parameter of said element, a phase modulator configured to modulate at a modulation frequency f mod , a phase of said laser radiation, by a modulation phase f modand adapted to vary said modulation phase, the phase-modulated radiation being injected into said cavity, a photodiode arranged to detect radiation transmitted or reflected by said cavity and configured to generate an electrical signal representative of the intensity of said detected radiation, a demodulation system adapted to demodulate said electrical signal at said modulation frequency f mod , for each gap d v i , by: o synchronously generating a first demodulated signal and a second demodulated signal representative of the demodulated electrical signal, respectively at a first demodulation phase φ dem,1 and to a second modulation phase φ dem,2 different from said first demodulation phase, such that φ dem,2 ≈ φ dem,1 ± k,with k ∈ [0; 2π], o filtering the first and second signals so as to retain only a continuous component of the first demodulated signal called error signal 1 V 1 and the second demodulated signal called the error signal 2 V 2 a processor connected to the modulation system and adapted to: o calculate a function 1 and a function 2 respectively representing an evolution of the error signal 1 and an evolution of the error signal 2 as a function of said difference dv for a given value of said first demodulation phase, o determine a value of the first demodulation phase called decoupling phase f dem, 1 = φdecfor which said function 1 or said function 2 has a plateau on a portion of values ​​of said deviation comprising 0, said function among X or Y having the plateau being called PL function, the function not having the plateau being called SP function, o determine, at said decoupling phase, a deviation called zero deviation for which said intensity of the light radiation detected by said photodiode reaches an extremum, and control said element on a value of said parameter making it possible to maintain this intensity extremum o modulate, at said decoupling phase, the phase of said laser radiation, with the phase modulator, by a periodic signal called additional at a modulation frequency called additional add in addition to that at the said modulation frequency f mod ;o minimize, for the SP function, an amplitude of a first harmonic of the additional signal by varying said second demodulation phase, this minimum being reached for a second demodulation phase called additional, and control said element on a value of said parameter making it possible to maintain this intensity extremum at said additional phase o determine, at said decoupling phase and at said zero difference, a value of said modulation phase for which said PL function is cancelled, and control said phase modulator in order to maintain the cancellation of said PL function.

[0032] According to a particular embodiment of the device of the invention, the element is said laser and said parameter is a direct voltage V dc,L laser power supply.

[0033] According to another particular embodiment of the device of the invention, the element is an acousto-optic modulator configured to vary said optical frequency of the laser radiation before it is phase modulated by said phase modulator, said parameter being an excitation frequency f AOM said acousto-optic modulator.

[0034] According to another particular embodiment of the device of the invention, the element is a piezoelectric translation stage on which a component of said cavity is fixed, said stage being adapted to vary a length of the cavity, said parameter being said length of the cavity.

[0035] According to a particular embodiment of the device of the invention compatible with all embodiments of the invention, the cavity is a ring cavity comprising an optical fiber and a first and a second coupling means configured to couple said radiation injected into said cavity with said optical fiber, the first and the second coupling means comprising a mirror or a fiber coupler.

[0036] According to a variant of the previous particular embodiment of the device of the invention, the device comprises: an optical separator adapted to separate the laser radiation into a first and a second optical path so as to inject said laser radiation into the cavity in a first and a second direction, the first optical path comprises the phase modulator and an optical circulator placed after the phase modulator, the second optical path comprises an additional phase modulator, and an additional optical circulator placed after the additional phase modulator, said optical circulator and said additional optical circulator being adapted to direct the injected laser radiation respectively in said second direction and in said first direction then reflected by the cavity towards an additional reflection photodiode and towards said photodiode, a first optical path of said first optical path and a second optical path of said second optical path between respectively the phase modulator and said cavity,and the additional phase modulator and said cavity being in guided optics. Brève description des figures :

[0037] Other characteristics, details and advantages of the invention will emerge from reading the description given with reference to the appended drawings given by way of example and which represent, respectively: There figure 1 , a schematic view of a laser servo device on a cavity of the prior art, The figure 2 , a schematic view of the polarization state of light as it propagates within a phase modulator, The figures 3A and 3B , a schematic view of an optical device according to the invention for achieving control of a laser on a laser cavity. figure 4 , a method according to the invention, The figures 5A and 5B , respectively an example of the PL and SP functions, before control, at the decoupling phase φ dec = -73.16° depending on dv / ISL THE figures 6A and 6B, respectively an example of the PL and SP functions, before control, at the decoupling phase φ dec = 92.44° as a function of dv / ISL, with the same cavity parameters as those used for the figures 5A and 5B , for operation in reflection, The figure 7 , an embodiment of the method of the figure 4 , There figure 8 , a device identical to that of the embodiment of the figure 3 except that the control is carried out on an acousto-optic modulator, The figure 9 , a device identical to that of the embodiment of the figure 3 except that the servo-control is carried out on a piezoelectric translation stage on which a component of the cavity is fixed, The figure 10 , an embodiment identical to the device of the figure 8 except that the cavity is a ring cavity and includes an optical fiber, The figure 11, an embodiment of the invention in which the device D comprises a passive ring resonant cavity C into which it is possible to inject the laser beam in a first direction, by a first optical path and in a second direction by a second optical path, The figure 12A And 12B , respectively an evolution of V ε 1 = ∑ k = 0 p Re b k cos φ dem + Im b k sin φ dem and of V ε 2 = ∑ k = 0 p − Re b k sin φ dem + Im b k cos φ dem as a function of Δ f , THE figures 13A and 13B , an evolution of V ε x = ∑ k = 0 p Re A k Re K k And V ε y = ∑ k = 0 p Re A k Im K k depending on f-mod / ISL.

[0038] References to figures, when identical, correspond to the same elements.

[0039] In the figures, unless otherwise indicated, the elements are not to scale. Detailed description:

[0040] THE figures 3A and 3Billustrate a schematic view of an optical device D according to the invention for carrying out a servocontrol of a laser L on a laser cavity C (in the figure a linear cavity but it can also be a ring cavity) while carrying out a correction of the RAM introduced by a phase modulator PM. The figure 3A illustrates a first variant in which the control and correction of RAM are carried out using a photodiode PD r detecting the radiation reflected by the cavity. The figure 3B illustrates a second variant in which the servoing and RAM correction are carried out using a photodiode PDt detecting the radiation transmitted by this cavity. This device is intended to implement the control method described in the figure 4 And 7 which we will detail later.

[0041] Structurally, the device D is close to that of the prior art presented in figure 1with one major exception, which is that the RAM correction is performed on a PDr or PDt photodiode, detecting radiation reflected or transmitted by the cavity C, which is the same one used to achieve the laser control on the cavity resonance. Thus, the device and the associated method allow the RAM to be compensated without the birefringence of the components placed after the phase modulator introducing errors in the RAM correction and in the laser control on the cavity. For clarity, the equations demonstrating that it is possible to control both the laser on the cavity and the RAM so that it does not introduce bias are given in the appendix section. It is also shown in this section that the laser control and the PM phase modulator control can be decoupled to cancel the RAM using error signals generated from a single PDr photodiode ( figure 3A ) or PDt ( figure 3B ).

[0042] In device D, the laser L generates a laser beam LL at an optical frequency f included in a resonance of the cavity C. This laser beam LL then passes through the phase modulator PM. The phase modulator applies a phase to the laser beam LL f mod which is modulated at a frequency f mod = ω mod / 2π. As seen previously, the phase modulator applies a phase ϕ mod = ϕ o , e + ϕ o , e dc + δ o , e sin ω mod t , along the ordinary axis “o” and the extraordinary axis “e”.

[0043] Moreover, as ϕ o , e + ϕ o , e DC = 2 πl λ n o , e − πl λd r o , e n o , e 3 V DC , it is possible to vary the modulation phase f mod by varying the DC voltage applied to the phase modulator.

[0044] The modulation frequency f-mod is applied by a local oscillator OL. The modulation depth M = δ e can take any value but in an assembly as described previously, it is typically 1.1 rad.Alternatively, according to another mode the modulation depth is 2.45 rad. Choosing a higher modulation depth allows the optical carrier to be suppressed. It is used in some ring cavity configurations to address the problem of backscattering in the cavity. For example, a passive resonant gyroscope consists of an optical cavity probed by two counterpropagating beams. Under the effect of rotation, the resonant frequencies will be modified differently (Sagnac effect) unless the backscattering is so strong that it locks the two optical carriers in frequency, preventing any speed measurement. One solution is to adapt the gyroscope to operate by suppressing the optical carriers.

[0045] Furthermore, in all embodiments of the invention, D comprises an element adapted to vary a gap dvbetween the optical frequency of the LL laser radiation and the central frequency fc , this deviation being controlled by a parameter of the element. Thus, this element makes it possible to achieve the control of the cavity resonance on the optical frequency of the laser or vice versa. More precisely, the element must be adapted so that the optical frequency f sweeps the resonance. In the embodiment of the figures 3A and 3B , it is the laser itself which is adapted to vary δv, via the direct voltage V dc,l powering the laser L. Alternatively, in the embodiment of the figures 8 and 9 , these are respectively the acousto-optic modulator AOM and a piezoelectric translation stage TS on which a cavity component is fixed which are the element allowing the difference δv to be varied.

[0046] After passing through the PM phase modulator, the phase-modulated laser beam passes through an optical circulator CO which may include a polarizer (not shown in figures 3A and 3B ) in order to fix the state of the polarization of the incident field on the cavity (or the polarizer can be placed between the circulator and the cavity). The presence of a polarizer before the cavity makes it possible to avoid introducing biases or instabilities due to the deformation of the resonances by the excitation of a bad polarization.

[0047] The phase-modulated radiation is then injected into cavity C. This cavity can be any type of resonant optical cavity known to those skilled in the art, linear or ring, fibered or not. It has a resonant mode centered at the central frequency fc with a line width Δf c . In the figures 3A and 3Ba first coupling means M1 and a second coupling means M2 are shown for coupling the laser beam LL to the cavity. These means can typically be mirrors or couplers in the case of a fiber cavity.

[0048] The device D comprises a single photodiode PDr or PDt arranged to detect respectively radiation transmitted or reflected by the cavity. The photodiodes PDr, PDt respectively generate a signal Sr, St representative of the intensity of the detected laser beam.

[0049] The device according to the invention and the associated methods implemented by the device operate with a single photodiode and therefore uses either the signal Sr detected by PDr, or the signal St detected by PDt. In the following the signal used is symbolized by St, Sr generated by the photodiode PDt, PDr.

[0050] The PDr photodiode is illuminated by the radiation reflected by C and then isolated by the optical circulator CO. Critically, for operation in reflection (i.e. control using the PDr photodiode), the sidebands must be as far as possible outside the cavity resonance, preferably but not necessarily f-mod >> Δf c / 2,. Otherwise, the behavior of the error signal at resonance is still linear as a function of δv but the slope is lower and the control may be less precise or even impossible.

[0051] The PDt photodiode is arranged so as to be illuminated by the transmitted radiation, for example by the coupling means M2. For operation in transmission (i.e. control using the PDt photodiode), the sidebands must be coupled into the cavity and then transmitted by the latter, i.e. included in the resonance, i.e. f-mod < Δf c . Otherwise, the sidebands would be too reflected by the cavity and the error signal would lose amplitude.

[0052] The device further comprises a demodulation system Dem adapted to demodulate the electrical signal at the modulation frequency f mod . For this, Dem includes an electrical signal mixer Mix connected to the local oscillator OL multiplying the signal generated by the photodiode by a signal of the form Acos(ω mod t + φ dem ).Importantly, the mixer Mix is ​​adapted to synchronously generate a first demodulated signal and a second demodulated signal. The first demodulated signal is representative of the electrical signal St, Sr demodulated at a first demodulation stage f dem, 1, or the signal St, Sr multiplied by Acos(ω mod t + φ dem , 1) . The second demodulated signal corresponds to the electrical signal St, Sr demodulated at a second modulation phase φ dem,2 different from said first demodulation phase, such that φ dem,2 ≈ φ dem,1 ± k, with k ∈ [0; 2π] or the signal St, Sr multiplied by Acos(ω mod t + φ dem, 1 ± k).

[0053] Preferably, the second demodulated signal corresponds to the electrical signal St, Sr demodulated at a second modulation phase φ dem,2 substantially in quadrature of said first demodulation phase, such that φ dem,2 ≈ φ dem,1 ±π / 2. By substantially in phase quadrature, we mean here that ϕ dem , 2 = ϕ dem , 1 ± π 2 ± 5 % . As will be explained later, the generation of two error signals with these quadrature phase demodulation phases is a possibility to ensure that by locking the two signals to zero, the laser and the cavity are properly locked and that the biases due to the RAM are eliminated. The choice of two quadrature demodulation phases is not mandatory and it will also be shown that a suitable choice of non-quadrature demodulation phases allows decoupling between the servo for the cancellation of the RAM and the servo of the laser.

[0054] In addition, the demodulation system comprises a low-pass filter LP adapted to filter the first and second signals so as to retain only a continuous component of the first demodulated signal called error signal 1 denoted V ε1 and of the second demodulated signal called error signal 2 denoted V ε2 .

[0055] Finally, the device comprises a UT processor configured to process the first and second error signals. This UT processor is further configured to perform the servo-control of the laser L on the cavity from the signals V ε1 and V ε2 (see below), while maintaining a supply voltage value allowing a zero deviation. dv = 0. Finally, the processor allows control of the PM phase modulator by maintaining a voltage value V dc,pm feeding PM allowing RAM to be canceled (see later).

[0056] There figure 4 illustrates a suitable method for controlling the cavity C on the laser L or vice versa and for compensating for the amplitude modulation introduced by a PM phase modulator. This method is implemented by the device of figures 3A, 3B And 6 à 9 .

[0057] This method comprises a first step A) consisting of generating, by the laser L, the laser radiation LL at the optical frequency f included in the resonance Δ fc of cavity C. Then, in a step B, the gap is varied dv between the optical frequency of LL and the central frequency fc of the resonance, so that the optical frequency sweeps the resonance. Thus, the gap dv takes a plurality of values ​​(δv 1 , ... δv k , ..., δv n ) and there are a plurality of gaps dv j such as δvj < 0 and there is a plurality of deviations dv i such as dv i > 0. As explained previously, the deviation is controlled by a parameter of an element of the device, this element being able to be the laser L, the acousto-optic modulator AOM or a piezoelectric translation stage TS.

[0058] For each deviation δv i , we carry out the following sub-steps i-iv: A step i) consisting of modulating, at a modulation frequency f mod , a phase of the LL laser radiation, by a modulation phase φ mod , with the phase modulator PM. The modulation frequency of PM is set by the local oscillator OL. A step ii) consisting of injecting the phase-modulated radiation LL into the cavity C. A step iii) consisting of detecting, by a photodiode PDt, PD r , a radiation reflected or transmitted by said cavity. The photodiode generates an electrical signal St, Sr representative of the intensity of the detected radiation.

[0059] Finally, a step iv) consisting of demodulating, using the modulation system Dem, the electrical signal St, Sr at the modulation frequency f-mod by synchronously generating a first demodulated signal and a second demodulated signal representative of the demodulated electrical signal, respectively at a first demodulation phase f dem,1 and a second modulation phase φ dem,2 . This pair of phases can a priori take any values ​​as long as they fulfill the condition set out in the appendix (condition A). To be sure of fulfilling this condition, we can choose the second one substantially in quadrature of said first demodulation phase, such that φ dem,2 ≈ φ dem,1 ± π / 2. Step iv further comprises a step of filtering, by the low-pass filter LP, the first and second demodulated signals so as to retain only a continuous component of the first demodulated signal. V 1 called error signal 1 and the second demodulated signal Vε 2 called error signal 2.

[0060] In a step C), the error signals V 1 and Vε 2 obtained for each gap dv i are processed by the UT processor. Step C) includes the calculation of a function f 1 and a function f 2 respectively representing an evolution of the error signal 1 and an evolution of the error signal 2 as a function of the deviation δv for a given value of φ dem,1 And φ dem,2 . So, Vε 1 = f 1 (δv) And Vε 2 = f 2 (δv). f 1 and f 2 can be calculated by any analytical methods known to those skilled in the art. f 1 and f 2 are the two error functions which will allow the RAM to be corrected while controlling the laser on the cavity (or vice versa) using a single PDr, PDt photodiode without inducing any error linked to the birefringence of components included in the device.

[0061] We now describe the method for adjusting the demodulation phase pair to decouple the servocontrols. The method of the figure 4 comprises a step D consisting of a repetition of steps B) and C) by varying the first demodulation phase f dem, 1 (it being understood that the variation of f dem, 1 leads to that of φ dem,2 which remains to f dem, 1 + k ) between each repetition until, for a value of the first demodulation phase called decoupling phase f dem, 1 = φ dec , function 1 or function 2 has a plateau on a portion P of values ​​of the deviation dv including 0. The correct demodulation phase is found when the plateau corresponds to a zero or near-zero voltage. If it is initially at a non-zero voltage, the plateau level must be brought back to zero voltage by applying a V DC voltage to modify Δφ dc<applied to the phase modulator. During this adjustment of the plateau level, the shape of the demodulated signal may be modified and we may observe that the signal no longer has a plateau. It is then necessary to act concomitantly on the demodulation phase to maintain the plateau. The P portion includes values ​​of δv which are all in the vicinity of 0. Subsequently, for greater clarity, we call the PL function the function among 1 or 2 presenting the plateau as a function of the deviation from resonance and we call the SP function the function not presenting the plateau. The SP function has a slope and a linear behavior around the resonance δv = 0 allowing a control of the cavity. By "presenting a plateau", we mean that, on the P portion, the derivative of the PL function is zero, i.e. df PL dδν 0 = 0 The LP function will be used to control the RAM because as shown in the equations presented in the appendices, this function is linear as a function of the RAM around the controlled value which cancels the effect of the RAM.

[0062] The PL signal, used to control the RAM, is thus decorrelated (in other words independent) of the δv deviations. That is to say that the signal applied to the phase modulator does not contain any component due to the fluctuations of the laser around the resonance of the cavity. Thus, the PL error signal whose function presents a plateau has two particular properties: close to a resonance it is independent of the laser frequency (presence of a plateau) but it depends linearly on the phase Φmod (see appendix and figures 13A, 13B ). The SP signal is used to control the deviation δv to zero.

[0063] THE figures 5A and 5B respectively present an example of the PL and SP functions, before control, at the decoupling phase φ decdepending on dv / ISL, with ISL the free spectral interval of cavity C. As a reminder, ISL = c / L cav with L car the optical length (which takes into account the index) of the cavity for a ring cavity and ISL = c / 2L cav for a linear cavity. In this example, without limitation, it is function 1, associated with the error signal Ve 1 , which presents a plateau at the decoupling phase φ dec = -73.16° and which therefore corresponds to the PL function. Function 2, associated with the error signal Ve 2 , therefore corresponds to the SP function. For a decoupling phase of φ dec = -73.16° + 90° = 16.84°, this would be the opposite. As a non-limiting example, the cavity C used to obtain these functions presented in figures 5A and 5B has a fineness of F = 12, a line width of Δf c = 1.2MHz and a free spectral interval ISL = 16MHz. The functions are obtained in a transmission operating mode (detection with PDt) with f mod = 400kHz. So we have well f-mod < Δf c / 2. There figure 5A allows us to observe that the PL function does indeed have a plateau on the P portion which is not zero because the RAM is not zero in this simulation. It is from this function that the cancellation of the RAM and the control to ensure that the cancellation of the RAM is maintained will be carried out. figure 5B allows us to observe that the SP function has a slope in the vicinity of dv / ISL on the P portion. It is this slope and the linear behavior of the SP function around the resonance which would make it possible to achieve the control of the optical frequency of the laser on the cavity (or vice versa).

[0064] We can therefore use this pair of demodulation phases. However, at this stage the SP signal is not decorrelated from the RAM.

[0065] THE figures 6A and 6Brespectively present an example of the PL and SP functions, before control, at the decoupling phase φ dec depending on dv / ISL, with the same cavity parameters C as those used for the figures 5A and 5B , for operation in reflection (detection on PDr). Here, without limitation, f mod = 4MHz. So we have well f-mod > Δf c / 2. Moreover, φ dec = 92.44° and it is function 2 which corresponds to the PL function while function 1 corresponds to the SP function. In these figures, on a portion of P values, we clearly observe the presence of a plateau for the PL function and a linear variation of the SP function.

[0066] After step D), the method of the figure 4 includes a step E consisting of, in the decoupling phase φ dec ,vary the gap δv so as to maximize the intensity of the light radiation detected by the photodiode PDt in transmission and minimize the intensity of the light radiation detected by the photodiode PDr in reflection. In both cases, we therefore seek to reach an extremum of the intensity. This maximum or minimum intensity is obtained for a gap dv almost zero because the RAM is not yet controlled, it introduces a bias on the SP signal. This signal can be zero while the laser is not well locked on a resonance of the cavity (this is what poses a problem). Alternatively, step E consists of varying δv until the SP function is cancelled.

[0067] After obtaining the cancellation of SP, step D) includes, in step E, a step of controlling the element (laser L, AOM or piezoelectric stage TS) on a value of the parameter making it possible to control the deviation dvso as to maintain the cancellation of the SP function and this quasi-maximum intensity on PDt and this quasi-minimum intensity on PDr. This control step is carried out by the UT processor via feedback electronics known per se, according to conventional control methods, for example, without being restrictive, with PI or PID feedback electronics. This type of feedback allowing the error signal to converge towards a desired value is well known in automation. Since the RAM is not yet controlled at this stage of the method, the cancellation of the SP function does not necessarily correspond to a resonance, that is to say that the transmitted intensity is not necessarily maximum or the reflected intensity minimum.The decoupling phase, that is to say the one for which we obtain an error function presenting a plateau, makes it possible to control the laser frequency on the resonance (or vice versa) by using the SP signal then the control of the PM in order to cancel the RAM (see further).

[0068] After step E, in order to make the signal SP independent of the RAM, in a step F, when the laser and the cavity are locked to each other, a periodic signal called the additional signal with a modulation frequency called the additional signal is applied to the phase modulator PM. f add , which adds up to the RF modulation at the modulation frequency f mod . That is, the PM is modulated by two signals simultaneously, the first at the frequency f add , and a second at the modulation frequency f mod . The frequency add of this additional signal must be much lower than the modulation frequency f monso that we can consider that we are modulating the DC bias of the phase modulator (i.e. the DC component). By "much lower" we mean here 10 times, preferably 20 times lower. The additional signal can be any periodic signal commonly used in signal processing by those skilled in the art, for example a triangular, square, sinusoidal or sawtooth signal. The correction signal applied to the laser or to the cavity, i.e. the one generated by the servo control via the SP function is also modulated because of the additional periodic signal.

[0069] The equations in the appendix show that we can find a so-called additional demodulation phase φ dem,1 = φ dec2 such that the SP function no longer depends on RAM.

[0070] To find this additional demodulation phase φ dec2 ,the demodulation phase is varied until the modulation on the SP function is attenuated as much as possible by the additional modulation on the phase modulator. In order to achieve this, the method comprises a step G, which consists of minimizing, on the SP signal, an amplitude of a first harmonic of the additional signal, by varying the first demodulation phase, this minimum being reached for the demodulation phase φ dec2 . Concretely, this minimization is carried out by a spectral analysis of the SP function (for example with a Fourier transform). After having found this additional demodulation phase φ dec2 , the additional signal is no longer applied to the PM phase modulator. A control is then again carried out on the element (laser L, AOM or piezoelectric stage TS) on a value of the parameter allowing this intensity extremum to be maintained, at the additional phase φdec2 .

[0071] So, using these two demodulation phases decoupling φ dec2 And φ dec respectively to lock the laser and the cavity and to control the RAM, we use signals which are decorrelated, that is to say that the laser control signal does not depend or depends little on the RAM and the RAM control signal does not depend or depends little on a deviation from the resonance.

[0072] Finally, the method of the figure 4 includes a final step H), consisting of the decoupling phase φ dec and at zero deviation, vary the first modulation phase f mod, 1 until canceling the PL function and maintaining the cancellation by a servocontrol of the PM phase modulator. Indeed, when the RAM is not controlled, the plateau around the resonance is not at a zero value as can be seen on the figure 4 . More precisely, step H) consists of varying the DC voltage V dc,msupply of the phase modulator up to a voltage value called RAM allowing to cancel the PL function and to control said phase modulator on the RAM voltage. Here again this control step is carried out by the UT processor via feedback electronics known per se, according to the classic control methods, for example, without being restrictive, with PI or PID feedback electronics. In practice, we seek, for the zero deviation and at the decoupling phase φ dec , V dc,m such as f PL (φ mod ) = 0 with f PL the PL function. Canceling the PL function allows the RAM to be canceled and allows for control of the DC voltage value. V dc,m power supply of the phase modulator even if there is a small deviation at resonance (see below), this is the interest of the decoupling phase φ dec This tension continues V dc,m is added to the PM modulation voltage from OL.

[0073] So, the method of the figure 4 allows, by choosing an appropriate demodulation phase, to control the PM phase modulator to cancel the RAM and control the optical frequency f of the LL laser beam on the resonance of the cavity (or vice versa) using a detection of the radiation transmitted or reflected by the cavity C on the same photodiode PDt, PDr. The problems linked to the difference in birefringence on the optical path which goes towards the RAM control photodiode and the useful optical path which goes towards the rest of the experiment are therefore resolved by the method of figure 4 since it is the same photodiode that is used to control the RAM. Similarly, unlike the prior art, it is no longer necessary to ensure that the demodulation phase used to generate the error signal that is used to lock the laser is the same as that used to cancel the RAM on the RAM control photodiode.

[0074] There figure 7 illustrates another embodiment of the method of the figure 4 . In this mode, all steps are identical to those of the method of the figure 4 , except that it comprises a step D1) after step D) and before step E) consisting of varying the modulation frequency f-mod until a slope of the SP function is maximum on the portion P of values, by repeating step D for each modulation frequency. This step is implemented by the UT processor. Concretely, this step D1 consists of maximizing the slope of the SP function by a variation of the modulation frequency f mod , while ensuring that we are still in the decoupling phase φ dec , that is to say by iteratively varying the demodulation phase f dem,1 so that the PL function always has a plateau on the P portion. Indeed, the variation of the modulation frequency induces variations on electronic delays which will modify the demodulation phase. Step D1 therefore makes it possible to define an optimal modulation frequency-decoupling phase pair (f mod,o ; φ dec,o )for which the slope of the SP function is maximized in its linear part, therefore for which the control is facilitated and the sensitivity of the frequency shift measurement with respect to the resonance is increased. Indeed, the slope is measured in V / Hz, and the greater this slope, the more sensitive we will be to a frequency measurement (Hz) on the error signal (V). By measuring the error signal (controlled to zero but in practice never equal to zero) and multiplying its value by the slope, we find our frequency shift δv. With a high slope it is easier to discriminate the measurement of a Hz. This method can only work with transmission detection, that is to say detection using the photodiode PDt and f mod < Δf c / 2.Indeed, it can be demonstrated that, in reflection, the modulation frequency allowing the maximum slope to be obtained for the SP function leads to the cancellation of the PL function. It is therefore impossible to achieve a control of the PM allowing the RAM to be cancelled.

[0075] THE figures 8 and 9 illustrate devices identical to that of the embodiment of the figure 3 except that in these devices, the control is carried out respectively, on an acousto-optic modulator AOM, and on a piezoelectric translation stage TS on which a component of the cavity C is fixed. The two operating variants in transmission and in reflection are illustrated here on the same diagram but it is understood that the device operates with only one of these two photodiodes.

[0076] In the embodiment of the figure 8, the device D comprises an acousto-optic modulator AOM adapted to shift and vary the optical frequency f of the laser beam LL passing through it so that it is included in a resonance of the cavity C. This modulator AOM is a component known to those skilled in the art. It is placed before the phase modulator PM and after the laser L and is excited at a frequency f AOM which is the parameter allowing to control the deviation δv. This device is suitable for implementing the methods of figure 4 And 7 , the UT processor being configured to control the AOM to a value of the excitation frequency f AOMallowing to reach the zero deviation by maximizing the intensity detected on the photodiode PDt,PDr or by canceling the SP function (step E)). This AOM can be advantageous, for example, in order to reach an optical frequency of the laser radiation which is difficult to achieve by a simple variation of the DC supply voltage of L. The AOM is frequency-controlled unlike the laser in the previous example. The AOM has a precision which can be sub-Hertzian (precision below Hz) which is very practical for applications of controlled frequency measurements.

[0077] In the embodiment of the figure 9 , the device D comprises a piezoelectric translation stage TS on which a component of the cavity C is fixed. This element can be for example the first coupling means M1 or the second coupling means M2 (for example laser mirrors). The translation stage TS is adapted to vary the length L carof the cavity by moving one of the coupling means relative to the other and thus varying the central frequency fc of the resonance (and therefore vary δv). Indeed, in the case of a linear cavity, there is a resonance at the central frequencies f c = k . c 2 L cav with k ∈ ℕ * . In the case of a ring cavity, there is a resonance at the central frequencies f c = k . c L cav , avec k ∈ ℕ * . In the embodiment of the figure 9 , the cavity length is therefore the parameter controlling δv. This device allows control of the cavity on an optical frequency f of the laser (which can for example be fixed) and is suitable for implementing the methods of figure 4 And 7 . In these methods, step E) is implemented by the UT processor which controls the translation stage to a value of the length L carof the cavity allowing to reach the zero gap by maximizing the intensity detected on the photodiode PDt,PDr or by canceling the SP function.

[0078] There figure 10 presents an embodiment identical to the device of the figure 8 except that the cavity C is a ring cavity and comprises an optical fiber FO. The first and second coupling means M1, M2 are configured to couple the LL radiation injected into the cavity with the optical fiber FO. In the embodiment of the figure 10 , these coupling means each comprise a mirror and a lens. Depending on the direction of injection of the LL beam into the cavity C, one means is adapted to focus the beam in the fiber and the other means is adapted to collimate the laser beam at the fiber output. In the illustration of the figure 10, and this as a non-limiting example, the direction of injection of the beam into the cavity is counterclockwise. The arrangement of the PDr and PDt photodiodes is dependent on the injection direction.

[0079] For a clockwise injection direction, the position of the injection path (the one that includes the laser, the AOM, the PM phase modulator) and the reflection detection path including the PDr photodiode is reversed compared to the arrangement of the figure 10 . Similarly, the position of the transmission detection path (the one which includes the PDt photodiode) must also be moved perpendicular to the beam represented as coming out of the optical fiber FO compared to the arrangement illustrated in figure 10 .

[0080] In the case where the FO fiber is polarization-maintaining, the device D may comprise (as illustrated in figure 10) a λ / 2 blade at the input and output of the fiber associated with a polarizer between M1 and M2 in order to fix the axis of the polarization of the beam within the cavity with that of the FO fiber.

[0081] Like the devices according to the invention presented previously, the photodiodes of the device of the figure 10 are connected to the demodulation system Dem and the signals from Dem are processed by the processor UT which carries out the control of the laser optical frequency on the cavity C and the control of the PM to cancel the RAM, according to the methods of the invention.

[0082] In the embodiment of the figure 10, the control of the laser optical frequency is carried out on the AOM by the UT processor by controlling the excitation frequency. Alternatively, according to another embodiment, the device does not include the AOM and the control by the UT processor is a control of the laser on the resonance frequency of the cavity (as in the device of the figure 3 ). A control of a piezoelectric translation stage on which one end of the optical fiber or one of the cavity mirrors is fixed is also possible (as in the device of the figure 9 ).

[0083] Alternatively, in the case where the cavity is completely fibered, the first M1 and the second coupling means M2 comprise a fibered coupler, for example a 1x2 coupler making it possible to extract (or couple, depending on the injection direction) a part of the LL radiation included in the cavity (or towards the cavity, depending on the injection direction).

[0084] Alternatively, according to another embodiment, the means M1 and M2 may comprise only one mirror if this is an off-axis parabola for example.

[0085] The advantage of using a ring cavity like that of the device of the figure 10is that it allows the realization of a passive resonant gyrometer. The measurement of the rotation speed in a resonant gyrometer is based on the fact that the free spectral interval of a resonant cavity is modified by the rotation of the latter (Sagnac effect). However, since there is no gain medium in the passive resonant cavity, the optical frequency of the LL laser beam has no reason to follow the so-called resonant frequency of the cavity (integer multiple of the free spectral interval), which will be modified by the rotation. This is why, in the case of a resonant gyrometer, it is necessary to have a servocontrol of the laser or an AOM allowing the resonant frequency of the cavity to be followed over time.

[0086] There figure 11presents an embodiment of the invention in which the device D comprises a passive ring resonant cavity C into which it is possible to inject the laser beam in a first direction (anti-clockwise), by a first optical path F and in a second direction (clockwise), by a second optical path F'.

[0087] In the embodiment of the figure 11 the device comprises an optical splitter LS adapted to separate the laser radiation LL into the first and second optical paths. This optical splitter LS can be a 50 / 50 splitter blade if the beam propagation is in free space or a 1x2 coupler if the beam propagation is fibered.

[0088] The first optical path comprises the acousto-optic modulator AOM, the phase modulator PM and the optical circulator CO placed after the phase modulator. The second optical path comprises an additional acousto-optic modulator AOM', an additional phase modulator PM' and an additional optical circulator CO' placed after the additional phase modulator.

[0089] The modulators PM and PM' modulate the phase at a frequency imposed by the local oscillator OL. Alternatively, according to another embodiment, each phase modulator is modulated at a frequency which may be different from the other, imposed by a local oscillator associated with this modulator. As explained previously, the presence of the modulators AOM and AOM' are optional in the device of the figure 11 .

[0090] The optical circulator CO is adapted to direct the injected laser radiation in the second direction (anticlockwise) then reflected by the cavity C towards an additional reflection photodiode PDr'. The additional optical circulator is adapted to direct the injected laser radiation in the first direction (clockwise) then reflected by the cavity C towards the photodiode PDr. These optical circulators are typically based on a Faraday rotator placed between two polarization separator cubes

[0091] The optical path of the laser beam after phase modulation is fibered into the two optical paths. In a non-limiting example, the optical fibers placed after the PM and PM' modulator are polarization-maintaining.

[0092] The device of the figure 11 includes a cavity C identical to that presented in figure 10 . In the device of the figure 11, the optical path of the first optical path and the second optical path between respectively the phase modulator PM and the cavity C, and the additional phase modulator PM' and the cavity C, is in guided optics, for example using a polarization-maintaining or non-polarization-maintaining optical fiber, or even a planar waveguide.

[0093] In addition, the device of the figure 11 includes an optional Ens set and an additional Ens' set, each consisting of a lens, a λ / 2 plate and a polarizer. The set is placed in front of the output of the fiber injecting the laser radiation from the first optical path. The additional set is placed in front of the output of the fiber injecting the laser radiation from the second optical path. These Ens, Ens' sets allow the beam to be collimated and a polarization to be imposed when it enters the cavity.

[0094] The device comprises a transmission photodiode PDt and an additional transmission photodiode PDt'. PDt is arranged to detect the radiation injected in the first direction into the cavity and then transmitted by the latter. PDt' is arranged to detect the radiation injected in the second direction into the cavity and then transmitted by the cavity. It is understood that, as in the figure 3A , according to a first variant of the invention the device comprises only the reflection photodiodes PDr, PDr' in order to carry out the control of the optical frequency and the cancellation of the RAM. Alternatively, as in the figure 3B , in a second variant, the device comprises only the transmission photodiodes PDt, PDt'.

[0095] As in the devices according to the invention presented previously, the photodiodes are connected to the demodulation system Dem and the signals from Dem are processed by the processor UT which carries out the control of the laser optical frequency on the cavity C and the control of the PM to cancel the RAM, according to the methods of the invention.

[0096] In the embodiment of the figure 11 , the control of the laser optical frequency is carried out by the UT processor on the laser L by controlling the DC supply voltage V dc,L . Alternatively, according to another embodiment, the device comprises an AOM and the control by the UT processor is a control of the AOM on the resonance frequency of the cavity (as in the device of the figure 8) or even a control of a piezoelectric translation stage on which one end of the optical fiber or one of the cavity mirrors is fixed (as in the device of the figure 9 ). Annex

[0097] This section demonstrates the possibility of managing RAM cancellation and laser control on the cavity (or cavity on the laser) from a single photodiode. Then, it is demonstrated that it is possible to decouple RAM cancellation and laser control on the cavity (or cavity on the laser) with two demodulation phases called decoupling phases using error signals generated from this single photodiode.

[0098] We first look for the mathematical expression of the error signal.

[0099] As demonstrated previously, the field E injof the LL radiation transmitted by the PM phase modulator and incident on the cavity C after crossing the birefringent components is of the form (see figure 2 and Equation 1): E inj = E 0 e iωt e i ϕ s + ϕ f 2 b ′ e i ϕ 3 + ϕ b ′ + a ′ e i ϕ 2 + ϕ a ′ γ ′ → = E 0 e iωt a ′ e iϕ o ′ e iδ o sinω mod t + b ′ e iϕ e ′ e iδ e sinω mod t , with ϕ 2 , 3 + ϕ b ′ , a ′ = ϕ o , e + ϕ o , e dc + δ o , e sinω mod t + ϕ b ′ , a ′ = ϕ o , e ′ + δ o , e sinω mod t And ϕ o , e ′ = ϕ o , e + ϕ o , e dc + ϕ a ′ , a ′ so by developing in harmonics of the modulation frequency: E inj = E 0 ∑ k = − ∞ + ∞ a ′ e iϕ o ′ J k δ o + b ′ e iϕ e ′ J k δ e e i ω + kω mod t = = E 0 e iωt ∑ k = − ∞ + ∞ j k e ikω mod t with jk = | a' |e iφ'o< J k (d o ) + | b' | e iφ'e< J k (δ e ) checking j -k = ( -1) k< jk

[0100] The reflected field, E r , or transmitted, And , by the cavity after modulation of the phase of the LL laser beam by PM at a frequency oh mod is written: E r , t = E 0 ∑ k = − ∞ + ∞ j k F ω + kω mod e i ω + kω mod t With F(w) the transfer function in transmission or reflection of the cavity.

[0101] The reflected intensity, I r , or transmitted, It , by the cavity is written: I r , t = I 0 ∑ k = − ∞ + ∞ ∑ p = − ∞ + ∞ j k j p ¯ F ω + kω mod F ω + pω mod ¯ e i k − p ω mod t , with l 0 the initial intensity at the input of the phase modulator.

[0102] If we only look at the frequency components (sidebands) at more or less the modulation frequency ±ω mod , we only keep the terms p=k+1 or p = k-1, i.e.: I r , t = I 0 ∑ k = 0 + ∞ − j k + 1 j k ¯ F ω − k + 1 ω mod F ω − kω mod ¯ + j k j k + 1 ¯ F ω + kω mod F ω + k + 1 ω mod ¯ e − iω mod t + I 0 ∑ k = 0 + ∞ − j k + 1 ¯ j k F ω − k + 1 ω mod ¯ F ω − kω mod + j k ¯ j k + 1 F ω + kω mod ¯ F ω + k + 1 ω mod e iω mod t

[0103] We ask: A k , k + 1 = j k j k + 1 ¯ = a ′ e iϕ o ′ J k δ o + b ′ e iϕ e ′ J k δ e a ′ e − iϕ o ′ J k + 1 δ o + b ′ e − iϕ e ′ J k + 1 δ e = a 2 J k δ 0 J k + 1 δ 0 + b 2 J k δ e J k + 1 δ e + a ′ b ′ e i ϕ e ′ − ϕ o ′ J k δ e J k + 1 δ 0 + e − i ϕ e ′ − ϕ o ′ J k δ o J k + 1 δ e = a 2 J k δ 0 J k + 1 δ 0 + b 2 J k δ e J k + 1 δ e + a b J k δ e J k + 1 δ 0 + J k δ o J k + 1 δ e cos Δ ϕ − i a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 sin Δ ϕ

[0104] With ΔΦ = ϕ e ′ − ϕ o ′ + ϕ a ′ − ϕ b ′

[0105] And

[0106] By demodulating this signal with the demodulation system by a signal of the form cos(ω mod t + ϕ dem ), then a low-pass filtering to keep only the continuous component we have an error signal V ε of the following form:

[0107] V ε = G tot 〈I r,t · sin (ω mod t + ϕ< dem)〉, with Gtot=GPD.Gmixer.Gfilter a term that groups together the gains of the different components, the photodiode, the signal mixer and the low-pass filter.

[0108] The general form of the error signal is (Equation 2) V ϵ = G tot I o ⋅ ∑ k = 0 + ∞ Re b k cos φ dem + Im b k sin φ dem

[0109] We take the expression of the error signal V ederived from Equation 2 by limiting the sum to the first p terms: V ε ΔΦ ν = G tot I o ⋅ ∑ k = 0 p Re b k cos φ dem + Im b k sin φ dem b k ΔΦ ν ≡ b k , 1 ΔΦ ν = A k , k + 1 ΔΦ ν F ν + kf mod F ν + k + 1 f mod ¯ − A k + 1 , k ΔΦ ν F ν − k + 1 f mod F ν − kf mod ¯ = A k ΔΦ ν F ν + kf mod F ν + k + 1 f mod ¯ − A k ΔΦ ν ¯ F ν − k + 1 f mod F ν − kf mod ¯

[0110] With A k ΔΦ ν ≡ A k , k + 1

[0111] In the following, we will take Great I o = 1 to simplify the notations. We simply use the property F r,t (v 0 - δv) = F r,t (v 0 + δv) (demonstrable) of the transmission function Ft or reflection F r in amplitude when dv is the frequency deviation from a resonance v 0 of cavity C.

[0112] b k Δ Φ , ν 0 = A k Δ Φ , ν 0 ⋅ F ν 0 + kf mod ⋅ F ν 0 + k + 1 f mod ¯ − A k ¯ Δ Φ , ν 0 ⋅ F ν 0 − k + 1 f mod ⋅ F ν 0 − kf mod ¯

[0113] b k Δ Φ , ν 0 = A k Δ Φ , ν 0 ⋅ F ν 0 + kf mod ⋅ F ν 0 − k + 1 f mod − A k ¯ Δ Φ , ν 0 ⋅ F ν 0 − k + 1 f mod ⋅ F ν 0 + kf mod

[0114] We then have (Equation 3): b k Δ Φ , ν 0 = A k Δ Φ , ν 0 − A k ¯ Δ Φ , ν 0 ⋅ F ν 0 + kf mod ⋅ F ν 0 − k + 1 f mod = 2 ilm A k Δ Φ , ν 0 ⋅ F ν 0 + kf mod ⋅ F ν 0 − k + 1 f mod

[0115] We want it to be zero at resonance even with RAM. So we have to cancel all the bk (v) so all the / m(A k (ΔΦ,v 0 )) what is possible with sin(ΔF) = 0 (see expression of A k ).

[0116] We note that the condition “to be at resonance (δv = v - v 0 = 0) with sin(ΔΦ) = 0 » is independent of the demodulation phase. If the coefficients bk (ΔΦ, v 0 ) are zero, the error signal V ε (ΔΦ, v) is zero for any demodulation phase.

[0117] It is now necessary to prove that there are error signals that allow us to control ΔΦ and lock the laser frequency to a cavity resonance. Indeed, there is no guarantee that we cannot have a situation in which we would have sin(ΔΦ) ≠ 0 and dv ≠ 0 and yet have bk (DV, v) zero, that is to say that unfortunately the defects of the phase modulator control would compensate for a locking error on a cavity resonance. In the following, we therefore show that being at resonance with sin(ΔΦ) = 0 is a necessary and sufficient condition to have bk (DF,u) null.

[0118] The aim of the calculation set out below is to show that only the condition “being at resonance (deviation δv = 0) and sin(ΔΦ) = 0 » allows to have bk (DF,u) zero. It is assumed that the laser frequency v 0 + δv and the phase ΔΦ 0 + δΔΦ can vary around their controlled values ​​δΔΦ = 0 and δv = 0, i.e. bk (sin(ΔΦ 0 ) = 0, vo ) = 0 (see Equation 3). We seek to know if it is possible to have, in the vicinity of this situation, other pairs of values ​​(ΔΦ, dv) such that all coefficients bk = 0 or zero too.

[0119] We recall that the coefficients F k depend only on the characteristics of the cavity and are independent of ΔΦ. We will use the properties of symmetries around a resonance frequency v 0 , valid in transmission as in reflection: F t ν 0 − δυ = F t ν 0 + δυ ¯ et dF t dυ ν 0 − δυ = − dF t dυ ν 0 + δυ ¯

[0120] We then have: b k ΔΦ + δ ΔΦ , ν 0 + δν = b k sin ΔΦ , ν 0 + δν db k dν ν 0 + δ ΔΦ db k d ΔΦ with bk (DF, v 0 ) (see Equation 3): b k ΔΦ ν 0 = A k ΔΦ ν 0 F ν 0 + kf mod F ν 0 + k + 1 f mod ¯ − A k ΔΦ ν 0 ¯ F ν 0 − k + 1 f mod F ν 0 − kf mod ¯ = − 2 Im A k ΔΦ ν 0 q k − ip k

[0121] By asking: p k + i q k = F ν 0 − k + 1 f mod F ν 0 − kf mod ¯ = F ν 0 + kf mod F ν 0 − k + 1 f mod

[0122] We then have: db k d ΔΦ = dA k ΔΦ ν 0 d ΔΦ F ν 0 + kf mod ⋅ F ν 0 + k + 1 f mod ¯ − d A k ΔΦ ν 0 ¯ d ΔΦ F ν 0 − k + 1 f mod ⋅ F ν 0 − kf mod ¯ = dA k ΔΦ ν 0 d ΔΦ F ν 0 + kf mod ⋅ F ν 0 − k + 1 f mod − d A k ΔΦ ν 0 ¯ d ΔΦ F ν 0 − k + 1 f mod ⋅ F ν 0 + kf mod = 2 i Im dA k ΔΦ ν 0 d ΔΦ F ν 0 + kf mod ⋅ F ν 0 − k + 1 f mod = − 2 i a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 p k + i q k cos ΔΦ ν 0 = 2 a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 q k − i p k cos ΔΦ ν 0

[0123] The term db k dν ν 0 is the sum of two terms, the first, db k dν ν 0 1 , describing the variation of db k due to a variation of the optical frequency v via the cavity response and the second, db k dν ν 0 2 , describing the variation of db k due to a variation of the optical frequency v via the response of the phase modulator or db k dν ν 0 = db k dν ν 0 1 + db k dν ν 0 2 db k dν ν 0 1 = A k ΔΦ ν 0 ⋅ d dν F ν 0 + kf mod ⋅ F ν 0 + k + 1 f mod ¯ − A k ΔΦ ν 0 ¯ ⋅ d dν F ν 0 − k + 1 f mod ⋅ F ν 0 − kf mod ¯ And db k dν ν 0 2 = dA k ΔΦ ν 0 d ΔΦ d ΔΦ dν ν 0 F ν 0 + kf mod ⋅ F ν 0 + k + 1 f mod ¯ − d A k ΔΦ ν 0 ¯ d ΔΦ d ΔΦ dν ν 0 F ν 0 − k + 1 f mod ⋅ F ν 0 − kf mod ¯ We have db k dν ν 0 1 = = A k ΔΦ dF ν 0 + kf mod dν F ν 0 + k + 1 f mod ¯ + A k ΔΦ F ν 0 + kf mod d F ν 0 + k + 1 f mod ¯ dν − A k ΔΦ ¯ dF v 0 − k + 1 f mod dν F ν 0 − kf mod ¯ − A k ΔΦ ¯ F ν 0 − k + 1 f mod d F v 0 − kf mod ¯ dν = A k ΔΦ dF ν 0 + kf mod dν F ν 0 − k + 1 f mod − A k ΔΦ F ν 0 + kf mod dF ν 0 − k + 1 f mod dν − A k ΔΦ ¯ dF ν 0 − k + 1 f mod dν F ν 0 + kf mod + A k ΔΦ ¯ F ν 0 − k + 1 f mod dF ν 0 + kf mod dν = A k ΔΦ + A k ΔΦ ¯ F ν 0 − k + 1 f mod dF ν 0 + kf mod dν − F ν 0 + kf mod dF ν 0 − k + 1 f mod dν = − 2 F ν 0 + kf mod dF dν ν 0 − k + 1 f mod − F ν 0 − k + 1 f mod dF dν ν 0 + kf mod Re A k ΔΦ And db k dν ν 0 2 = = dA k ΔΦ d ΔΦ d ΔΦ dν F ν 0 + kf mod ⋅ F ν 0 + k + 1 f mod ¯ − d A k ΔΦ ¯ d ΔΦ d ΔΦ dν F ν 0 − k + 1 f mod ⋅ F ν 0 − kf mod ¯ = dA k ΔΦ d ΔΦ F ν 0 + kf mod ⋅ F ν 0 − k + 1 f mod − d A k ΔΦ ¯ d ΔΦ F ν 0 − k + 1 f mod ⋅ F ν 0 + kf mod d ΔΦ dν = 2 i Im dA k ΔΦ d ΔΦ F ν 0 + kf mod ⋅ F ν 0 − k + 1 f mod d ΔΦ dν

[0124] To obtain this result, we made the assumption that the modulation depths δ o , e = − πl λd r o , e n o , e 3 V RF evolved little with the difference δv since d δ o , e dν = − πl cd r o , e n o , e 3 V RF = δ o , e ν and that the δ o,e are at best a few radians. This implies that d δ o , e dν is of the order of 10 -14< .

[0125] So we have db k dν ν 0 2 = 2 a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 q k − i p k cos Δ Φ d Δ Φ dν = db k d Δ Φ ⋅ d Δ Φ dν

[0126] The term db k dν ν 0 r é sonance 2 can be neglected compared to db k dν ν 0 r é sonance 1 .Indeed, ΔΦ is close to (2k + 1) π / 2 and being a phase term, it is proportional to the optical frequency v or ΔΦ = αν et d ΔΦ dν ΔΦ 0 = α = ΔΦ 0 ν . If we assume that we propagate over 10 m and that the birefringence is such that we accumulate a phase shift of 2π every λ then we accumulate, at 1.55 µm, 4·10 6< rads or d ΔΦ dν ΔΦ 0 ≈ 2 ⋅ 10 − 8 while the multiplicative term 2 |a||b|[J k (δ 0 )J k+1 (δ e )-J k (δ e )J k+1 (δ 0 )] [qk -ip k ] can be bounded above by taking a = b = 1 , J k δ o J k + 1 δ e = J 0 2 0 = 1 , J k δ e J k + 1 δ 0 = 0 , F ν 0 + kf mod ⋅ F(v 0 - (k + 1)f mod ) = 1 or 2 |a|| b | [J k (d o )J k+ 1 (δ e ) -J k (δ e )J k+ 1 (d 0 )] qk - ipk ≤ 2

[0127] We therefore have (Equation 4): db k = δν db k dν + δ ΔΦ db k d ΔΦ = 2 a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 q k − i p k cos ΔΦ δ ΔΦ − 2 K k ⋅ Re A k ⋅ δν using pk + iqk = F(v 0 + kf mod )F(v 0 - (k + 1 )f mod ), K k = F ν 0 + kf mod dF dν ν 0 − k + 1 f mod − F ν 0 − k + 1 f mod dF dν ν 0 + kf mod

[0128] Which gives (Equation 5): b k = − 2 Im A k ΔΦ ν 0 q k − ip k − 2 K k ⋅ Re A k ⋅ δν

[0129] By demodulating the error signal at a first demodulation phase ϕ dem 1 and a second demodulation phase ϕ dem2as in the methods of the invention, the error signal 1 is obtained V 1 and the error signal 2 Vε 2 such that: V ε 1 = G tot I o ⋅ ∑ k = 0 + ∞ Re b k cos φ demod 1 + Im b k sin φ demod 1 V ε 2 = G tot I o ⋅ ∑ k = 0 + ∞ Re b k cos φ demod 2 + Im b k sin φ demod 2

[0130] We want to know if it is possible to have a couple (δΔΦ, dv) of non-zero values ​​leading to V ϵ 1 = cos φ demod 1 ∑ k = 0 p Re b k + sin φ demod 1 ∑ k = 0 p Im b k = 0 V ϵ 2 = cos φ demod 2 ∑ k = 0 p Re b k + sin φ demod 2 ∑ k = 0 p Im b k = 0 If the demodulation phases are chosen so that the system determinant, det1 = cos( f demod1 ) sin( f demod2 ) - sin( f demod1 ) cos( f demod2), is not zero (condition A) then the only solution is: ∑ k = 0 p Re b k = ∑ k = 0 p Re b k ΔΦ 0 + δ ΔΦ , ν 0 + δν = 0 ∑ k = 0 p Im b k = ∑ k = 0 p Im b k ΔΦ 0 + δ ΔΦ , ν 0 + δν = 0 soit ∑ k = 0 p Re db k = 0 ∑ k = 0 p Im db k = 0 because ∑ k = 0 p Re b k ΔΦ 0 ν 0 = 0 et ∑ k = 0 p Im b k ΔΦ 0 ν 0 = 0 .

[0131] We are therefore looking for solutions to ∑ k = 0 p Re db k = 2 δ ΔΦ cos ΔΦ 0 ∑ k = 0 p a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 q k − 2 δν ∑ k = 0 p Re A k Re K k = 0 ∑ k = 0 p Im db k = − 2 δ ΔΦ cos ΔΦ 0 ∑ k = 0 p a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 p k − 2 δν ∑ k = 0 p Re A k Im K k = 0 δ ΔΦ cos ΔΦ 0 a b ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 q k − δν ∑ k = 0 p Re A k Re K k = 0 δ ΔΦ cos ΔΦ 0 a b ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 p k + δν ∑ k = 0 p Re A k Im K k = 0

[0132] The only solution of this system is ( dd Φ = 0 ,δv = 0) because the determinant of the system is non-zero. We have therefore proven that (δΔΦ = 0 , δv = 0) is then the only solution reached when the servocontrols have converged.

[0133] Among the choice of a pair of demodulation phases, the choice of two quadrature phases guarantees that the determinant det1 is not zero since it is equal to 1.

[0134] In the following, we will keep the choice of two quadrature phases because it allows us to derive analytical formulas. We will show that it is possible to decorrelate the control signals in transient conditions. We show that we can achieve the condition dv = 0 regardless of the condition sin(ΔF)= 0 which actually makes the system insensitive to RAM. We could therefore assume that there is no point in removing RAM. The problem is that this decoupling demodulation phase can vary under the influence of variations in experimental conditions. For example, this demodulation phase can vary if the optical and electrical delays vary, for example, due to temperature. It is therefore necessary to remove RAM to overcome this. We will also show that we can achieve the condition sin(ΔF) = 0 regardless of the condition dv = 0 this is so that the signal applied to control the RAM does not contain any component due to laser drift.

[0135] We now seek to have one of the two error signals Vé 1 or Vε 2 which changes little (ideally not at all) when the RAM changes around the controlled value sin(ΔF)= 0. Thus, if the RAM control shifts from this ideal value, for example with a malfunction of its servo, at first order this does not affect the value of this error signal which we will show can be used to lock the laser and the cavity. Thus, this error signal has two particular properties: close to a value such that sin(ΔF) = 0, it is independent of a fluctuation of the RAM but it depends linearly on the frequency difference δv between the laser and a resonance of the cavity.

[0136] We are therefore looking for two demodulation phases ϕ demod And ϕ demod + π / 2 such as the signal used to control the cavity, V e 1 or V ε2 , does not depend or depends little on a variation of the RAM, when the laser is in resonance with the cavity (vo = pc / L opt ,ΔΦ = (k + 1)π), soit dV ε 1 , 2 d ΔΦ = 0 soit dV ε 1 d ΔΦ = d d ΔΦ cos φ demod ∑ k = 0 p Re b k + sin φ demod ∑ k = 0 p Im b k = 0 Or dV ε 2 d ΔΦ = d d ΔΦ − sin φ demod ∑ k = 0 p Re b k + cos φ demod ∑ k = 0 p Im b k = 0 either cos φ demod d d ΔΦ ∑ k = 0 p Re b k + sin φ demod d d ΔΦ ∑ k = 0 p Im b k = 0 Or − sin φ demod d d ΔΦ ∑ k = 0 p Re b k + cos φ demod d d ΔΦ ∑ k = 0 p Im b k = 0

[0137] As (see Equation 5) b k sin ΔΦ 0 = 0 + δ ΔΦ , ν 0 résonance = − 2 a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod δ ΔΦ i

[0138] The equation (C) leads to (as the coefficients J k (δ o ), J k+1 (δ e ), J k (δ e ), J k+1 (δ 0 ) are real) cos φ demod Im ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod = sin φ demod Re ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod

[0139] The desired demodulation phase is therefore given by φ demod 1 = arg ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod + p π

[0140] Equation (D) leads to − sin φ demod Im ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod = cos φ demod Re ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod

[0141] The desired demodulation phase is therefore given by tan φ demod = − Re ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod Im ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod either φ demod 2 = arg ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod + π 2 + p π = φ demod 1 + π 2

[0142] We see that this demodulation phase depends only on the modulation depth and the transfer function of the cavity. This demodulation phase corresponds to the additional phase φ dec2 mentioned above, and which is such that the SP function no longer depends on RAM.

[0143] We now check that it is possible to use one of the two error signals to lock the laser and the cavity. We recall that the error signals used are (forgetting the term G tot I o ): V ε 1 ≡ ∑ k = 0 + ∞ Re db k cos φ demod + Im db k sin φ demod with a demodulation phase cos φ demod d d ΔΦ ∑ k = 0 p Re b k + sin φ demod d d ΔΦ ∑ k = 0 p Im b k = 0 And V ε 2 ≡ ∑ k = 0 + ∞ Re db k cos φ demod + π / 2 + Im db k sin φ demod + π / 2 with a demodulation phase − sin φ demod d d ΔΦ ∑ k = 0 p Re b k + cos φ demod d d ΔΦ ∑ k = 0 p Im b k = 0 with the expression of bk presented in Equation 5, and (see above) A k = a 2 J k δ 0 J k + 1 δ 0 + b 2 J k δ e J k + 1 δ e + a b J k δ e J k + 1 δ 0 + J k δ o J k + 1 δ e cos ΔΦ − i a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 sin ΔΦ and with (see Equation 4): db k = δν db k dν + δ ΔΦ db k d ΔΦ = 2 a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 q k − i p k cos ΔΦ δ ΔΦ − 2 K k ⋅ Re A k ⋅ δν either V ε 1 = 2 a b δ ΔΦ ∑ k = 0 p q k cos φ demod − sin φ demod p k J k δ 0 J k + 1 δ e − J k δ e J k + 1 δ 0 − 2 δν cos φ demod ∑ k = 0 p Re K k ⋅ Re A k + sin φ demod ∑ k = 0 p Im K k ⋅ Re A k

[0144] As the demodulation phase is chosen to cancel the first term, we have a linear error signal allowing us to control dv to zero and independent of ΔΦ. Similarly V ε 2 = − 2 a b δ ΔΦ ∑ k = 0 p q k sin φ demod + p k cos φ demod J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 − 2 δν − sin φ demod ∑ k = 0 p Re K k ⋅ Re A k + cos φ demod ∑ k = 0 p Im K k ⋅ Re A k

[0145] We see that there are two phases of decoupling in phase quadrature from each other such that at the decoupling phase and at zero deviation, the plateau level of the PL function is zero when the RAM is zero. Thus, we verify that (sDF) = 0,δ v =0) is the only solution reached when the servocontrols have converged. The PL function is in fact zero for any value of ΔΦ while it is linear as a function of δv and can be used to lock the laser and the cavity.

[0146] On reflection, we have φ demod 1 = arg ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 F ν 0 + kf mod F ν 0 − k + 1 f mod + p π

[0147] As the sidebands are clearly outside the cavity resonance, they are almost totally reflected and we have either F ν 0 + kf mod ≈ − F ν 0 − k + 1 f mod either φ demod 1 ≈ − arg ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 = 0

[0148] As p k + i q k = F ν 0 + kf mod F ν 0 − k + 1 f mod ≈ − 1 + 0 i SO V ε 1 ≈ 2 a b δ ΔΦ sin 0 ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 ≈ 0

[0149] Likewise, φ demod 2 = φ demod + π 2 , V ε 2 ≈ 2 a b δ ΔΦ cos φ demod ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 ≈ 2 a b δ ΔΦ cos π 2 ∑ k = 0 p J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 ≈ 0

[0150] In reflection, decoupling exists but it leads to a zero or at least very low slope for the error signals. In order to compensate for this low slope, preferably in reflection, in step B iv) we demodulate at 3x the modulation frequency f mod .

[0151] We then seek to have one of the two error signals V 1 or Vε 2 which changes little (ideally not at all) when the laser frequency changes around the resonance. Thus, if the laser shifts relative to the resonance, for example with a malfunction of its servo, to the first order this does not affect the value of this error signal which we will show can be used to servo sin(ΔΦ). Thus, this error signal has two particular properties: close to a resonance it is independent of the laser frequency but it depends linearly on the phase ΔΦ.

[0152] We are therefore looking for a demodulation phase such that a plateau, defined by a portion of the P value of the deviation over which the error signal changes little (ideally not at all, i.e. tangent to the horizontal curve) when the laser frequency changes, exists around a resonance for one of the two error signals Vé x Or Vé y .

[0153] We are therefore looking for two demodulation phases ϕ dem 1 and ϕ dem,2 = P dem 1 + π / 2 such that on a portion P with a deviation value close to 0 d dν cos φ dem ∑ k = 0 p Re b k + sin φ dem ∑ k = 0 p Im b k = 0 Or d dν − sin φ dem ∑ k = 0 p Re b k + cos φ dem ∑ k = 0 p Im b k = 0

[0154] Either A cos φ dem d dν ∑ k = 0 p Re b k + sin φ dem d dν ∑ k = 0 p Im b k = 0 Or B − sin φ dem d dν ∑ k = 0 p Re b k + cos φ dem d dν ∑ k = 0 p Im b k =

[0155] Using the expression of bk presented in Equation 5 and as presented previously, K k = F ν 0 + kf mod dF dν ν 0 − k + 1 f mod − F ν 0 − k + 1 f mod dF dν ν 0 + kf mod the previous equation (A) is written: sin φ dem Re ∑ k = 0 p Re A k K k + cos φ dem Im ∑ k = 0 p Re A k K k = 0 tan φ dem = − tan ∑ k = 0 p Re A k K k φ dem 3 = arg ∑ k = 0 p Re A k K k + π 2 + pπ

[0156] Similarly, equation B leads to a demodulation phase allowing the decoupling given by: tan φ dem = tan ∑ k = 0 p Re A k K k

[0157] Either, φ dem 4 = arg ∑ k = 0 p Re A k K k + pπ = φ dem 3 − π 2

[0158] We see that there are therefore two phases of decoupling in quadrature of each other (see figures 5A, 5B And 6A, 6B ). The PL function represents the evolution of the error signal showing a plateau as a function of dv which is used to control RAM.

[0159] We now check that it is possible to use the error signal giving the PL function presenting a plateau in order to control the phase ΔΦ via the DC voltage V dc,PM applied to the phase modulator.

[0160] We remind you that the signals used are: V ε 1 ≡ ∑ k = 0 + ∞ Re b k cos φ demod + Im b k sin φ demod With a decoupling phase cos φ dem ∑ k = 0 p Re A k Re K k + sin φ dem ∑ k = 0 p Re A k Im K k = 0 Or V ε 2 ≡ ∑ k = 0 + ∞ Re b k cos φ dem + π / 2 + Im b k sin φ dem + π / 2 With a demodulation phase − sin φ demod ∑ k = 0 p Re A k Re K k + cos φ demod ∑ k = 0 p Re A k Im K k = 0

[0161] With the expression of bk presented in Equation 5, and (see above) A k = a 2 J k δ 0 J k + 1 δ 0 + b 2 J k δ e J k + 1 δ e + a b J k δ e J k + 1 δ 0 + J k δ o J k + 1 δ e cos ΔΦ − i a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 sin ΔΦ

[0162] And with (see Equation 4): db k = δν db k dν + δ ΔΦ db k d ΔΦ = 2 a b J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 q k − i p k cos ΔΦ δ ΔΦ − 2 K k ⋅ Re A k ⋅ δν

[0163] We have: V ε 1 = 2 a b δ ΔΦ ∑ k = 0 p q k cos φ dem − sin φ dem p k J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 − 2 δν cos φ dem ∑ k = 0 p Re K k ⋅ Re A k + sin φ dem ∑ k = 0 p Im K k ⋅ Re A k

[0164] However, the decoupling phase allows the second term to be cancelled. As desired, we therefore obtain a linear error signal allowing ΔΦ to be controlled to zero and independent of δv.

[0165] THE figure 12A And 12B respectively represent the evolution of V ε1 = ∑ k = 0 p Re b k cos φ dem + Im b k sin φ dem and of V ε 2 = ∑ k = 0 p − Re b k sin φ dem + lm(bk) cos(ϕ dem ) as a function of Δ f These evolutions were obtained with the same cavity parameters as those used for the figures 5A to 6B . Here, the modulation frequency is f mod = 0.4MHz and δv = 80kHz and a transmission operation, that is to say the case in which the cancellation of the RAM and the control is done with the PDt photodiode. We see that despite a deviation at the resonance of 80 kHz, the error signal V εy is canceled for ΔΦ = 0 and remains linear around which allows to ensure a control canceling the RAM. We note that if the other decoupling phase had been used, it would be the error signal V ε1 which would be canceled for ΔΦ = 0 and which would allow to realize the control.

[0166] In reflection, the interest is limited because the sidebands being almost reflected we have dF dν ν 0 − k + 1 f mod ≈ dF dν ν 0 + kf mod ≈ 0 et F ν 0 + kf mod ≈ − F ν 0 − k + 1 f either K k = F ν 0 + kf mod dF dν ν 0 − k + 1 f mod − F ν 0 − k + 1 f mod dF dν ν 0 + kf mod ≈ 0 SO φ dem 3 = arg ∑ k = 0 p Re A k K k + π 2 + pπ ≈ π 2 et φ dem 4 ≈ 0 As the sidebands are clearly outside the cavity resonance, they are almost totally reflected and we have pk + iqk = F(v 0 + kf mod )F(v 0 - (k + 1) f-mod ) ≈ -1 + 0 i SO V ε 1 = 2 a b δ ΔΦ cos π 2 ∑ k = 0 p q k J k δ o J k + 1 δ e − J k δ e J k + 1 δ 0 = 0 Likewise Vé 2 = 0.

[0167] In reflection, decoupling exists but it leads to a zero or at least very low slope for the error signals. In order to compensate for this low slope, preferably in reflection, in step B iv) we demodulate at 3x the modulation frequency f mod .

[0168] Thus, using one of the two demodulation phases ϕ dem 1 or ϕ dem2 , and another demodulation phase ϕ dem3 Or ϕ dem4 , it is possible to independently enslave sin(ΔΦ) = 0 and δv = 0. We are then sure that the system converges towards sin(ΔΦ) =0 (RAM is zero) by controlling the DC offset of the phase modulator using an error signal that varies linearly around the point ΔΦ = pπ and contains no component coming from a (small) drift of the laser with respect to the cavity and dv = 0 using an error signal to control the laser frequency which will not contain any component from the RAM and which varies linearly around the point ( dv = 0).

[0169] Furthermore, if we demodulate in phase and in quadrature then the error signals are proportional to V ε x = ∑ k = 0 p Re A k Re K k And V ε y = ∑ k = 0 p Re A k Im K k . It can be shown that there exists a modulation frequency f-mod such as : ∑ k = 0 p Re A k Re K k = 0 ou ∑ k = 0 p Re A k Im K k = 0 .

[0170] THE figures 13A and 13B present an evolution of V ε x = ∑ k = 0 p Re A k Re K k And V ε y = ∑ k = 0 p Re A k Im K k depending on f-mod / ISL. These evolutions were obtained with the same cavity parameters as those used for the figures 5A to 6B, in transmission. In Figure 14A, we notice that the condition V ε x = ∑ k = 0 p Re A k Re K k = 0 can be achieved for f mod = 0.0313102 ISL. We also see in Figure 14B that V ε y = ∑ k = 0 p Re A k Im K k = 0 , which is the slope of the error signal to control the optical frequency of the laser on the cavity -or the reverse- (see Equation 4), passes through a maximum at this value of f mod . Thus, for the modulation frequency canceling one of the error signals, the other error signal has a maximum slope.

Claims

1. A method for the servo control of an optical device comprising a cavity (C) exhibiting a resonance around a central frequency fc, a laser (L) and a phase modulator (PM), said method being adapted to servo-control said cavity (C) to said laser (L) or vice versa and to compensate for an amplitude modulation introduced by said phase modulator (PM), said method comprising the following steps: A. generating, through said laser (L), laser radiation (LL) at an optical frequency f within said resonance, B. varying a difference δv between said optical frequency of said laser radiation and said central frequency, such that said optical frequency scans said resonance, said difference being controlled by a parameter of an element of said device, and for each difference δvi i. modulating, at a modulation frequency fmod, a phase of said laser radiation, through a modulation phase φmod, with the phase modulator (PM), ii. injecting the phase-modulated radiation into said cavity (C), iii. detecting, through a photodiode (PDt, PDr), radiation reflected or transmitted by said cavity and generating an electrical signal (St, Sr) representative of the intensity of said detected radiation, iv. demodulating said electrical signal at said modulation frequency fmod by synchronously generating a first demodulated signal and a second demodulated signal representative of the demodulated electrical signal, respectively at a first demodulation phase φdem,1 and at a second modulation phase φdem,2, with φdem,2 ≈ φdem,1 ± k, where k ∈ [0; 2π] is different from said first phase, and by filtering the first and the second signal so as to retain only a DC component of the first demodulated signal Vε1, called error signal 1, and of the second demodulated signal Vε2, called error signal 2, C. computing a function 1 f2 and a function 2 f2 respectively representing an evolution in the error signal 1 and an evolution in the error signal 2 as a function of said difference for a given value of said first demodulation phase, D. repeating steps B) and C) by varying said first demodulation phase between each repetition until, for a value of the first demodulation phase, called decoupling phase φdem,1 = φdec, said function 1 or said function 2 has a plateau on a portion of values of said difference comprising 0, said function out of 1 or 2 having the plateau being called PL function, the function not having the plateau being called SP function; E. at said decoupling phase, varying said difference so as to observe a peak, for what is called a zero difference, said intensity of the light radiation detected by said photodiode, and servo-controlling said element to a value of said parameter that makes it possible to maintain this intensity peak, F. at said decoupling phase, modulating the phase of said laser radiation with the phase modulator (PM), through a so-called additional periodic signal at a so-called additional modulation frequency fadd in addition to that at said modulation frequency fmod ; G. minimising, for the SP function, an amplitude of a first harmonic of the additional signal, by varying said first demodulation phase, this minimum being reached for a so-called additional first demodulation phase φdem,1 = φdec2, and, at said additional phase, servo-controlling said element to a value of said parameter that makes it possible to maintain this intensity peak, H. at said decoupling phase φdem,1 = φdec and at said zero difference, varying said first modulation phase until the PL function is cancelled out, and maintaining the cancelling out of said PL function by servo-controlling said phase modulator.

2. The method according to claim 1, wherein step H) consists in varying a voltage Vdc,pm of said phase modulator up to a so-called RAM voltage value that makes it possible to cancel out the PL function and servo-controlling said phase modulator to said RAM voltage.

3. The method according to claim 1 or 2, wherein, in step iii), said photodiode detects the radiation transmitted by said cavity, said method comprising a step D1), after step D) and before step E), consisting in varying said modulation frequency until a slope of said SP function is at a maximum on said portion of values, by repeating step D for each modulation frequency.

4. The method according to any one of the preceding claims, wherein the additional frequency fadd is 10 times lower than the modulation frequency fmod.

5. An optical device (D) comprising a cavity (C) exhibiting a resonance around a central frequency fc and a laser (L) adapted to generate laser radiation (LL) at an optical frequency f within said resonance, said device comprising: - an element adapted to vary a difference δv between said optical frequency of said laser radiation and said central frequency, such that said optical frequency scans said resonance, said difference being controlled by a parameter of said element, - a phase modulator (PM) configured to modulate, at a modulation frequency fmod, a phase of said laser radiation, through a modulation phase φmod, and adapted to vary said modulation phase, the phase-modulated radiation being injected into said cavity (C), - a photodiode (PDt, PDr) arranged to detect radiation transmitted or reflected by said cavity (C) and configured to generate an electrical signal representative of the intensity of said detected radiation, - a demodulation system (Dem) adapted to demodulate said electrical signal at said modulation frequency fmod, for each difference δvi, by: ∘ synchronously generating a first demodulated signal and a second demodulated signal representative of the demodulated electrical signal, respectively at a first demodulation phase φdem,1 and at a second modulation phase φdem,2 different from said first demodulation phase, such that φdem,2 ≈ φdem,1 ± k, with k ∈ [0; 2π], ∘ filtering the first and the second signal so as to retain only a DC component of the first demodulated signal, called error signal 1 Vε1, and of the second demodulated signal, called error signal 2 Vε2, - a processor (UT) connected to the modulation system and adapted to: ∘ compute a function 1 and a function 2 respectively representing an evolution in the error signal 1 and an evolution in the error signal 2 as a function of said difference δv for a given value of said first demodulation phase, ∘ determine a value of the first demodulation phase, called decoupling phase φdem,1 = φdec, for which said function 1 or said function 2 has a plateau on a portion of values of said difference comprising 0, said function out of X or Y having the plateau being called PL function, the function not having the plateau being called SP function, ∘ determine, at said decoupling phase, a difference, called zero difference, for which said intensity of the light radiation detected by said photodiode reaches a peak, and servo-control said element (E) to a value of said parameter that makes it possible to maintain this intensity peak, ∘ modulate, at said decoupling phase, the phase of said laser radiation, with the phase modulator (PM), through a so-called additional periodic signal at a so-called additional modulation frequency fadd in addition to that at said modulation frequency fmod; ∘ minimise, for the SP function, an amplitude of a first harmonic of the additional signal, by varying said second demodulation phase, this minimum being reached for a so-called additional second demodulation phase, and servo-control said element to a value of said parameter that makes it possible to maintain this intensity peak at said additional phase, ∘ determine, at said decoupling phase and at said zero difference, a value of said modulation phase for which said PL function is cancelled out, and servo-control said phase modulator (PM) in order to maintain the cancelling out of said PL function.

6. The device according to the preceding claim, wherein said element is said laser (L) and said parameter is a DC voltage Vdc,L for supply to the laser.

7. The device according to claim 5, wherein the element is an acousto-optic modulator (AOM) configured to vary said optical frequency of the laser radiation before it is phase-modulated by said phase modulator, said parameter being an excitation frequency fAOM of said acousto-optic modulator.

8. The device according to claim 5, wherein the element is a piezoelectric translation stage to which a component of said cavity is fixed, said stage being adapted to vary a length of the cavity, said parameter being said length of the cavity.

9. The device according to one of claims 5 to 8, wherein the cavity (C) is a ring cavity comprising an optical fibre (FO) and first and second coupling means (M1, M2) configured to couple said radiation injected into said cavity with said optical fibre, the first and second coupling means comprising a mirror or a fibre coupler.

10. The device according to claim 9, comprising: - an optical splitter (LS) adapted to split the laser radiation into a first and a second optical channel (F, F') so as to inject said laser radiation into the cavity in a first and a second direction, - the first optical channel comprises the phase modulator (PM) and an optical circulator (CO) placed after the phase modulator, - the second optical channel comprises an additional phase modulator (PM'), and an additional optical circulator (CO') placed after the additional phase modulator, - said optical circulator (CO) and said additional optical circulator being adapted to direct the laser radiation injected respectively in said second direction and in said first direction and then reflected by the cavity (C) toward an additional reflection photodiode (PDr') and toward said photodiode (PDr), - a first optical path of said first optical channel and an second optical path said second optical channel between the phase modulator and said cavity respectively, and the additional phase modulator and said cavity having a guided-optic configuration.