## 1.

## Introduction

The fabrication of a navigation-grade miniature optical gyroscope has been the aim of an old quest. Indeed, the possible realization of a miniature optical gyroscope integrated on an optical chip and having a bias stability better than $1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}/\mathrm{h}$ could have a strong impact on the medium/high performance gyroscope market, currently dominated by the well-established ring laser gyroscope^{1} or interferometric fiber optical gyroscope.^{2} The resonant miniature optical gyroscope (RMOG), based on a waveguide-type ring resonator, is an attracting approach, where the reduction of the optical path length is compensated for by the cavity $Q$ factor, as for other passive resonant devices.^{3} Such RMOG becomes more and more realistic due to the progress of the photonic integrated circuit (PIC) and whispering gallery mode resonator (WGMR) technologies. PIC technology, on the one hand, offers the potential of cost reduction through a collective manufacturing process like the semiconductor industry and the generalization of the multiwafer-run capability implemented by several foundries. There are now several platforms assisting end-users to design and manufacture PICs.^{4}^{,}^{5} PIC development achieved a remarkable improvement in the losses,^{6}^{,}^{7} heterogeneous integration of active indium phosphide gain sections or photodiodes with passive silicon or silicon nitride circuits,^{8}^{,}^{9} and packaging.^{10} On the other hand, WGMRs are not always fully compatible with collective manufacturing as they need to be diamond polished or they need to be heterogeneously reported on a substrate or coupled to a tapered fiber to inject and extract light. However, some of the best performances of miniature cavities and gyroscopes were demonstrated^{11} with this approach.

In such a gyro architecture, the passive optical cavity is probed by two coherent counterpropagating waves, and the rotation rate is retrieved by monitoring the modifications of the resonant frequencies of the clockwise (CW) and counterclockwise (CCW) beams circulating inside the cavity. From the literature,^{3} it is known that the partially reduced sensitivity of the RMOG could also be compensated for by a higher laser incident power, to reduce shot noise. However, it is also known that a difference in the intensities of the CW and CCW beams induces, via the Kerr effect, a nonreciprocal index difference^{2} resulting in a bias in the gyroscope response.^{12} Increasing the incident laser power then leads to the necessity of a tighter control of this difference, which becomes increasingly challenging. In this paper, we conduct an analysis that shows that it is possible, under realistic assumptions, to find an expression of the minimum cavity diameter and the maximum incident power that the cavity should be probed with to meet a given requirement on both the angular random walk (ARW) and bias stability of such a gyroscope. Our analysis takes into account (i) the PIC and WGMR technology performances in terms of propagation losses and mode size and (ii) the difference of the counter propagating beam intensities. We then use this simple model to assess the feasibility of a RMOG for navigation applications.

The paper is organized as follows. In Sec. 2, we introduce our cavity model. In Sec. 3, we use this model to derive the necessary minimum cavity diameter and the maximum incident power allowed to meet a set of navigation grade performances. As these values depend on the ring waveguide propagation losses, the Kerr effect coefficient, and the mode size, we conduct in Sec. 4, a parametric analysis using numbers from state-of-the-art PIC and WGMR technology to assess the feasibility of low (tactical) and medium performance RMOGs. We give some perspectives in terms of gyro development in Sec. 5. These sections are supported by five appendices, where we give the details of some calculations.

## 2.

## Cavity Model

The rotation sensing ring cavity of diameter $D$ is schematized in Fig. 1. We suppose that the cavity has only one coupler.^{13}

To maintain the laser at resonance, we implement a Pound–Drever–Hall (PDH) locking scheme.^{14}^{,}^{15} We use the notations of Stokes et al.^{16} by introducing the intensity losses ${\gamma}_{0}$ and the intensity coupling coefficient $\kappa $ of the coupler, which relate the coupler output and input intensities through the following relations:

In Appendix A, we use these definitions to derive the expressions of the intracavity field and of the field reflected by the cavity. This leads to the following full-width at half-maximum for the cavity resonance dip:

## (3)

$$\mathrm{\Delta}{\nu}_{\mathrm{FWHM}}=\frac{1-{\mathrm{e}}^{-\alpha L/2}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa}}{\pi}\mathrm{\Delta}{\nu}_{\mathrm{FSR}}\equiv \frac{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}{\mathcal{F}},$$The noise properties of the PDH locking scheme can be optimized by maximizing the slope of the error signal at resonance. We show in Appendix B that this optimum is obtained when the cavity obeys the so-called critical coupling condition,^{18} for which the resonance dip goes down to zero reflection, and is obtained by equating the coupling factor with the internal losses of the resonator:

As shown in Appendix B, optimizing the slope of the PDH error signal leads to a result different from the optimization of the cavity finesse because the slope depends on both the finesse and the contrast of the resonances. In the case where the condition of Eq. (4) is satisfied, Eq. (3) becomes

## (5)

$$\frac{\mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}=\frac{1-{\mathrm{e}}^{-\alpha L}(1-{\gamma}_{0})}{\pi}.$$To enhance the sensitivity of our gyroscope, it is important that the losses be small. In this case, we can use the following approximation:

and suppose that the coupler losses ${\gamma}_{0}$ are much smaller than the propagation losses, leading to Using Eqs. (4), (5), and (7), this leads to## (8)

$$\frac{\mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}=\frac{1}{\mathcal{F}}\simeq \frac{\alpha L}{\pi}=\alpha D,$$For a cavity under critical coupling conditions, we show in Appendix A that the intracavity intensity ${|{E}_{4}|}^{2}$ at resonance is given by

## (9)

$${\left(\frac{{|{E}_{4}|}^{2}}{{|{E}_{1}|}^{2}}\right)}_{\text{resonance}}=\frac{1-{\gamma}_{0}}{1-(1-{\gamma}_{0}){\mathrm{e}}^{-\alpha L}}.$$In the low loss approximation, we can use Eq. (7) to obtain:

## (10)

$${\left(\frac{{|{E}_{4}|}^{2}}{{|{E}_{1}|}^{2}}\right)}_{\text{resonance}}\simeq \frac{1}{\kappa}\simeq \frac{1}{\alpha \pi D}.$$This expression, already obtained in Ref. 19, stresses the direct link between the intracavity intensity and the propagation losses for a high finesse cavity in the critical coupling regime.

## 3.

## Shot Noise and Kerr Effect Related Performance Limits

We now focus on the ultimate achievable performances of the RMOG, assuming in the following sections that the only limitations are due to the shot noise and the Kerr effect. All other sources of noises, such as the laser source or electronic noises, or limitations, for instance, from Rayleigh backscattering,^{20} or polarization noise,^{21}^{,}^{22} are supposed to be mitigated. Of course, in a real development, these should be conveniently addressed, which is not a task that should be underestimated. We will come back to Brillouin scattering at the end of Sec. 4 to show that it is negligible with the materials that we consider.

## 3.1.

### Shot Noise Limit

The smallest measurable angular velocity in a time $\tau $ depends on the slope of the PDH servolocking slope, the Sagnac effect scale factor,^{23} and the shot noise level. Its expression is derived in Appendix C and reads

## (11)

$$\delta {\dot{\theta}}_{\mathrm{SNL}}=\frac{{n}_{0}\mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}{D}\sqrt{\frac{\lambda hc}{2\chi \tau {P}_{0}}},$$## 3.2.

### Kerr Effect Induced Bias

We then focus on the bias instability resulting from the Kerr effect. We show in Appendix D that the Kerr effect introduces a rotation rate bias given by

## (12)

$${\dot{\theta}}_{\mathrm{Kerr}}=\frac{c{n}_{2}\mathcal{F}}{2\pi \sigma D}\mathrm{\Delta}{P}_{0},$$^{12}

^{,}

^{21}

The bias of Eq. (12) should be maintained below the maximum bias ${\dot{\theta}}_{\mathrm{bias}}$ required by the gyro specifications, thus requiring a certain level of control of $\mathrm{\Delta}{P}_{0}$. For example, for a silica cavity (${n}_{2}=2.7\times {10}^{-20}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}$)^{24} of finesse $\mathcal{F}=40$, diameter $D=25\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, and mode section area $\sigma =33\text{\hspace{0.17em}\hspace{0.17em}}{\mu \mathrm{m}}^{2}$, Eq. (12) leads to a bias ${\dot{\theta}}_{\mathrm{Kerr}}/\mathrm{\Delta}{P}_{0}=4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}/\mathrm{s}/\mathrm{mW}$, in fair agreement with the measured value of $5.3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}/\mathrm{s}/\mathrm{mW}$ reported in Ref. 25, after having taken into account the factor of 2 coming from the fact that only one half of the incident power is at the carrier frequency that is resonant with the cavity.

## 3.3.

### Derivation of the Cavity Design Guidelines

Equations (11) and (12) predict the ultimate performances of a RMOG depending on design parameters. Let us shortly summarize the hypothesis along which these expressions were obtained: (i) The cavity exhibits low losses. The numerical results given later will prove that this must be the case to reach the desired performances. (ii) The coupler losses ${\gamma}_{0}$ are negligible compared with the propagation losses $\alpha L$. This is a reasonable assumption for state-of-the-art PICs couplers and waveguides.^{6} However, this hypothesis may be slightly too optimistic for WGMRs. (iii) A PDH locking scheme is implemented and the phase modulation amplitude is set at its optimum^{15} ($\beta =1.08\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}$). (iv) The cavity is set at the critical coupling to optimize the PDH slope (see Appendix B). Although this is not a critical parameter for our analysis, this can be a tricky issue in terms of manufacturing as the propagation losses might not be totally reproducible or predictable. However, it was demonstrated in Refs. 26 and 27 that thermally controlled variable couplers can be integrated on a PIC, allowing postfabrication tuning of the coupling efficiency, at the expense of potential drifts or errors if an active control is necessary. (v) The detector efficiency $\chi $ is equal to 1, and the shot noise limit is reached.

Equations (11) and (12) contain different kinds of parameters. Some of them, such as $\lambda $, $c$, and $h$, are physical constants that have fixed values. Some of them depend on the degree of maturity of the chosen technology. This is the case of $\alpha $, ${n}_{2}$, $\sigma $, and ${n}_{0}$, which depend on the chosen PIC or WGMR technology, and of the degree to which one is able to balance the two couterpropagating intracavity powers, which we will parametrize by introducing the following notation:

where $\mathrm{\Delta}P/P$ and $\mathrm{\Delta}{P}_{0}/{P}_{0}$ are the relative unbalance of the intracavity and incident powers, respectively. The smallest achievable value of the parameter $\xi $ depends on the efforts put into the active control of $\mathrm{\Delta}{P}_{0}$ through relevant electronics controls. Finally, the two remaining parameters in Eqs. (11) and (12) are $D$ and ${P}_{0}$, which are the true design parameters of the gyro.Consequently, our aim here is to derive the values of the design parameters $D$ and ${P}_{0}$ to achieve a given performance ARW and ${\dot{\theta}}_{\mathrm{bias}}$, taking into account the parameters $\alpha $, ${n}_{2}$, $\sigma $, and ${n}_{0}$ of the chosen technology. By combining Eqs. (11) and (12) with Eqs. (8) and (10), we obtain the following constraints on ${P}_{0}$ and $D$:

## (14)

$$\frac{{P}_{0}}{{D}^{2}}<\frac{2\pi}{c}\frac{\sigma \alpha}{{n}_{2}}\frac{1}{\xi}{\dot{\theta}}_{\mathrm{bias}},$$One can see that these relations impose contradictory constraints on the input optical power ${P}_{0}$. In particular, for a fixed value of $D$, Eq. (14) imposes a “maximum” value of ${P}_{0}$ to limit the bias instabilities, which scales like $\alpha $, while Eq. (15) imposes a “minimum” value for ${P}_{0}$ to reach the required ARW performance, which scales like ${\alpha}^{2}$.

## 4.

## Applications to Tactical and Medium Performance Gyroscopes

A tactical grade (i.e., low precision navigation grade) gyroscope usually has the following performance requirements:

## (16)

$$\mathrm{ARW}\le 0.1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}/\sqrt{\mathrm{h}}=2.9\times {10}^{-5}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}/\sqrt{\mathrm{s}},$$## (17)

$${\dot{\theta}}_{\mathrm{bias}}\le 1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}/\mathrm{h}=4.8\times {10}^{-6}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}/\mathrm{s},$$## (18)

$$\mathrm{ARW}\le 0.01\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}/\sqrt{\mathrm{h}}=2.9\times {10}^{-6}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}/\sqrt{\mathrm{s}},$$## (19)

$${\dot{\theta}}_{\mathrm{bias}}\le 0.1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}/\mathrm{h}=4.8\times {10}^{-7}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}/\mathrm{s}.$$Given those specifications, the question now is to decide which PIC or WGMR technology is the most favorable to build a RMOG that could meet them. Table 1 summarizes the propagation losses and nonlinear index of four different PIC technologies and one WGMR technology, namely silicon-on-insulator (SoI, see Refs. 28 and 29), indium phosphide (InP, see Refs. 3031.32.–33), silicon nitride (SiN, see Refs. 34 and 35), silicon-chip-based monolithic silica resonators (${\mathrm{SiO}}_{2}$, see Refs. 7, 36, and 37), and ${\mathrm{CaF}}_{2}$ WGMRs, which are heterogeneously reported on a micro-optical chip.^{11}

## Table 1

Propagation losses α and nonlinear index of refraction n2 for five different PIC materials.

Technology | α | n2 |
---|---|---|

SoI | $2.7\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dB}/\mathrm{m}$ (Ref. 38) | $5\times {10}^{-18}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}$ (Ref. 39) |

InP | $0.35\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dB}/\mathrm{cm}$ (Refs. 32 and 33) | ${10}^{-16}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}$ (Ref. 40) |

SiN | $0.32\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dB}/\mathrm{m}$ (Ref. 6) | $2.4\times {10}^{-19}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}$ (Ref. 41) |

${\mathrm{SiO}}_{2}$ | $0.11\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dB}/\mathrm{m}$ (Ref. 7) | $2.7\times {10}^{-20}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}$ (Ref. 24) |

${\mathrm{CaF}}_{2}$ | $0.0016\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dB}/\mathrm{m}$ (Ref. 11) | $3.6\times {10}^{-20}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}$ (Ref. 42) |

Note also that we suppose that the performances reported in Table 1 were all obtained for single-mode waveguides, which may not always be the case.

From Table 1, the most promising technologies seem to be the ones based on SiN, ${\mathrm{SiO}}_{2}$, and ${\mathrm{CaF}}_{2}$ for two reasons. First, these materials are the one with which the lowest propagation losses were demonstrated. Second, their nonlinear indices of refraction ${n}_{2}$ are smaller than the other materials mentioned in Table 1. Actually, SiN is even more favorable than what can be seen in this table because its “effective nonlinear index of refraction” can be made quite close to the one of silica. Indeed, because the contrast between the refractive indices of SiN and ${\mathrm{SiO}}_{2}$, which is used as the substrate for the PIC, is pretty small, the mode profile can be tailored to be only weakly confined so that the field mainly propagates inside ${\mathrm{SiO}}_{2}$. Actually, as can be seen in Ref. 6, the confinement factor $\eta $, defined as the fraction of the mode power that propagates in the central SiN core,^{43} is only 0.03, meaning that the effective nonlinear index is approximately given by

## (20)

$${n}_{2,\mathrm{eff}}=\eta {n}_{2}(\mathrm{SiN})+(1-\eta ){n}_{2}({\mathrm{SiO}}_{2})\approx 3.4\times {10}^{-20}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}.$$Moreover, with such a low confinement, the mode area $\sigma $ is equal to $33\text{\hspace{0.17em}\hspace{0.17em}}{\mu \mathrm{m}}^{2}$ (see Ref. 6), thus further decreasing the bias induced by the Kerr effect. The mode diameter in ${\mathrm{SiO}}_{2}$ resonators is taken to be equal to $37\text{\hspace{0.17em}\hspace{0.17em}}{\mu \mathrm{m}}^{2}$ (see Ref. 37). The ${\mathrm{CaF}}_{2}$ WGMR technology benefits from an even larger mode diameter, namely $\sigma =190\text{\hspace{0.17em}\hspace{0.17em}}{\mu \mathrm{m}}^{2}$, which we deduced from Ref. 42.

From Table 1, SoI could also have been a possible candidate, provided some improvement on the propagation losses would be achievable. However, because the index of refraction contrast between the silicon layer and the ${\mathrm{SiO}}_{2}$ substrate is very high, the mode remains mainly confined inside the silicon layer, meaning that the effective Kerr effect is close to the one of the bulk material reported in Table 1, i.e., 100 times larger than that of ${\mathrm{SiO}}_{2}$ and 10 times that of SiN (for further comparison between SoI and SiN, see, for instance, Ref. 44). Concerning InP, it is also worth mentioning that other nonlinear effects, such as the two-photon absorption, could become detrimental to the gyro performance even before the Kerr effect itself becomes a problem.

## 4.1.

### Tactical Grade Gyroscope

With the figures that we have obtained for SiN and ${\mathrm{SiO}}_{2}$ PICs and ${\mathrm{CaF}}_{2}$ WGMRs, respectively, Figs. 2(a)–2(c) show the range of the parameters $D$ and ${P}_{0}$ for which the limit is compatible with tactical performances [see Eqs. (16) and (17)] for four values of the maximum power imbalance $\xi $. These areas are obtained using the conditions given by Eqs. (14) and (15).

The first thing we notice by comparing Figs. 2(a) and 2(b) is that the results are quite similar for the two considered PIC materials. Looking more closely into details, one can see that for $\xi ={10}^{-2}$, the diameter of the gyro must be larger than 6 or 4.5 cm in the cases of SiN and ${\mathrm{SiO}}_{2}$, respectively. These are no longer really miniature dimensions. Besides, with such a value of $\xi $, the optical power must be reduced to a few $\mu \mathrm{W}$ to mitigate the Kerr effect bias. Only a power imbalance control as good as $\xi ={10}^{-3}$, which is achievable with a servo-loop control, can permit a reduction of $D$ below 4 cm in the case of SiN and below 3 cm in the case of ${\mathrm{SiO}}_{2}$, with an optical power of the order of $10\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{W}$. A power imbalance control as good as $\xi ={10}^{-4}$ could permit a decrease in the cavity diameter down to 2 and 1.5 cm for SiN and ${\mathrm{SiO}}_{2}$, respectively.

From Fig. 2(c), one could *a priori* believe that ${\mathrm{CaF}}_{2}$ WGMR can achieve tactical grade performances with smaller dimensions than PIC technologies. However, this impression must be mitigated by several observations: (i) the level of power needed to control the Kerr effect induced bias becomes so low, in the range of $1\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{W}$, that detection noise problems may become an issue; (ii) the calculations of Fig. 2(c) have been performed by assuming critical coupling, which is far from being the case in real implementations like the one of Ref. 11, where coupling losses are 16 times larger than internal losses; (iii) the tapered fiber coupling technique used for WGMR probably induces coupler losses, that are neglected here, which will further degrade the performance.

As a partial conclusion, Fig. 2 stresses the fact that one really needs to take into account the role of the Kerr effect, and not only the shot noise limit, in the achievable ultimate performance of the gyro. They also show that a control of the power imbalance between the two counter propagating curves is unavoidable.

## 4.2.

### Medium Performance Gyroscope

The situation is even worse in the case of a medium performance gyroscope, i.e., with the performance specifications given by Eqs. (18) and (19). As can be seen in Fig. 3, an active control of the power imbalance as good as $\xi ={10}^{-5}$ can only allow a reduction in the minimum cavity diameter down to 6 cm in the case of SiN and to 5 cm in the case of ${\mathrm{SiO}}_{2}$. From Fig. 3(c), it seems possible to reduced this diameter down to 3 or 4 cm in the case of ${\mathrm{CaF}}_{2}$ WGMRs. But the same discussion on the validity of the hypothesis as in the discussion on tactical performance gyroscopes applies here also.

With SiN and ${\mathrm{SiO}}_{2}$ PIC technologies, respectively, a tremendously precise power control ($\xi ={10}^{-6}$) is necessary to obtain the desired performance for $D=5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{cm}$ and $D=3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{cm}$, as can be seen from Figs. 3(a) and 3(b). Figure 3(c) suggests that this diameter could only be slightly reduced by the use of a ${\mathrm{CaF}}_{2}$ WGMR.

With the materials that we have chosen, and with the typical diameters and optical powers that we consider, one can check that Brillouin scattering is negligible, as shown in Appendix E.

## 4.3.

### Discussion

The main conclusion of this simple analysis is that the Kerr effect deeply impacts the RMOG design as it rapidly limits the level of power allowed to probe the cavity. For such a given power limit, the only way to improve the sensitivity at the shot noise limit is to increase the cavity diameter, leading to cavity dimensions that become comparable with other optical gyro technologies. Even for the tactical grade gyroscope, which has the less stringent performances, relative power differences well below the percent level are required. With a power imbalance level in the ${10}^{-3}$ to ${10}^{-4}$ range, a miniature tactical grade gyroscope seems achievable with a ring cavity diameter in the few cm range. Making a medium performance RMOG seems much more problematic because the relative difference of power should be below ${10}^{-5}$, not even mentioning the fact that the cavity should remain monomode with a single polarization or the manufacturing and cost issues associated with such a large diameter resonator.

In view of resonant gyro applications, the PIC or WGMR design should consequently focus not only on the propagation losses but also on the mode area $\sigma $ to decrease the effective Kerr effect. Even with a reduction of the Kerr effect, operation of a truly miniature ring with a diameter of a few cm implies a mandatory active control of the intensities. Indeed, such a level of power imbalance control (${10}^{-5}$ for a 3-cm diameter tactical grade diameter RMOG for instance) seems impossible to achieve in a passive way. In this respect, the demonstration in Ref. 45 of relative power imbalance actively reduced down to $2.5\times {10}^{-5}$ opens the way to such controls. The intensity modulation scheme proposed in Ref. 46 is also well adapted to the small cavities that we consider here, contrary to the one described in Ref. 12.

To obtain the above results, we have assumed that the coupler losses, ${\gamma}_{0}$, are negligible compared with the propagation losses $\alpha L$ and the coupler transmission $\kappa $. This assumption should be reassessed for the very small coupling values in the case of PIC resonators and may be far from being valid in the case of WGMRs, as discussed above. This would make things worse if this hypothesis is no longer true. Indeed, some extra coupler losses would lead to an increase of the cavity linewidth and smaller PDH slope, as can be seen from Eqs. (40)–(42), in Appendix B. The effect of these extra losses on the PDH slope should then be compensated for by increasing the incident power and/or the gyroscope resonator diameter. We have also assumed that the cavity is exactly tuned to the critical coupling regime. This not only makes all the calculations simpler, allowing us to derive formulas that can be used as simple design rules of a RMOG (and actually for any resonant optical gyroscope) but also optimizes the PDH slope. Thus, should the cavity coupling be set at a different regime, this would also lead to a need to increase the incident power or/and the gyroscope size to meet the desired performance limits.

Moreover, we assumed throughout this paper that the phase modulation depth was optimal for the PDH locking (i.e., 1.08 rad). However, it was shown^{20} that a modulation depth of 2.4 rad should be used to suppress the carrier and thus reduce the effect of the backscattering that we mentioned as a noise to be addressed. However, this modulation scheme leads to a somehow less steep error signal because the dominant terms are now proportional to ${J}_{1}(\beta ){J}_{2}(\beta )$, as shown in Ref. 47. A way to keep a modulation depth equal 1.08 rad while addressing the backscattering issue could be to interrogate the cavity with three different frequencies separated by an integer number of cavity free-spectral ranges, as proposed in Ref. 48.

## 5.

## Conclusion

In conclusion, we have performed a simple analysis of a PIC-based or WGMR-based RMOG with few realistic assumptions, namely critical coupling, high-Q cavity, negligible coupling losses, and PDH locking scheme driven with the optimal modulation depth. We have derived design rules to calculate the minimum gyroscope diameter and the maximum power to probe it, taking the shot noise limit and the Kerr-effect induced bias into account. From this, we conclude that the Kerr effect has a deep impact on the design of the gyroscope and that the cavity $Q$-factor is not the unique parameter governing the gyro performance. To meet the bias stability requirement, the Kerr effect puts a limit on the acceptable difference between the powers of the two counter-propagating probing beams. With a fixed given relative power imbalance, this drastically limits the maximum input power. It then becomes necessary to increase the gyroscope dimensions to fulfill the ARW requirement.

More precisely, we conclude that even the goal of building a cm-diameter-scale tactical grade gyroscope (i.e., a relatively low performance grade in terms of inertial navigation) already puts a strong constraint on the power balance, making it very challenging to build and raising cost-related issues. Assuming some of the best demonstrated PIC or WGMR technologies so far, reaching the performances of such a gyroscope requires a mitigation of the Kerr effect, for instance, with an active control of the counter propagating beam intensities. Second, for the same reasons, important improvements on the technology, such as a strong decrease in the losses, would be necessary to make a medium grade RMOG feasible. A more radical approach could be to get rid of the Kerr effect itself by making the light propagate mainly in air as with hollow-core fibers,^{49} by exploiting circular hollow core waveguides,^{50}^{,}^{51} by adapting the slot-waveguide approach^{52}^{,}^{53} to air core, or by pushing to its limits the wedge resonator geometry^{7}^{,}^{36} so that the sharp part of the waveguide is so thin that the mode confinement drastically decreases and the field mainly propagates in air. Another approach to reduce the Kerr effect limitation would be to identify new modulation solutions adapted to small cavities.

Finally, let us mention that another approach,^{54} based on an active Brillouin resonator, seems also promising to achieve a performant miniature optical gyroscope.

## Appendices

## Appendix A:

### Cavity Resonance and Linewidth

In this appendix, we derive the expression of the field reflected by and inside the cavity, to obtain Eqs. (3)–(10). Inside the cavity (see Fig. 1), the fields ${E}_{2}$ and ${E}_{4}$ are related by

where the round-trip phase shift is given by## (22)

$$\mathrm{\varphi}=\frac{2\pi \nu {n}_{0}L}{c}=\frac{\omega}{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}},$$## (23)

$${E}_{4}=\frac{i\sqrt{1-{\gamma}_{0}}\sqrt{\kappa}}{1-{\mathrm{e}}^{-\alpha L/2}{\mathrm{e}}^{-\mathrm{i}\mathrm{\varphi}}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa}}{E}_{1}.$$## (24)

$${\left(\frac{|{E}_{4}{|}^{2}}{|{E}_{1}{|}^{2}}\right)}_{\text{resonance}}=\frac{\kappa (1-{\gamma}_{0})}{{(1-{\mathrm{e}}^{-\alpha L/2}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa})}^{2}}.$$In the critical coupling regime given by Eq. (4), we retrieve the expression of Eq. (9).

The field reflected by the cavity is obtained by injecting Eqs. (21) and (23) into Eq. (1), leading to

## (25)

$${E}_{3}=\sqrt{1-{\gamma}_{0}}\left[\frac{\sqrt{1-\kappa}-{\mathrm{e}}^{-\alpha L/2}{\mathrm{e}}^{-i\mathrm{\varphi}}\sqrt{1-{\gamma}_{0}}}{1-{\mathrm{e}}^{-\alpha L/2}{\mathrm{e}}^{-i\mathrm{\varphi}}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa}}\right]{E}_{1}.$$This leads to the following expression for the reflected intensity:

## (26)

$$\frac{{|{E}_{3}|}^{2}}{{|{E}_{1}|}^{2}}=(1-{\gamma}_{0})[1-\kappa \frac{1-(1-{\gamma}_{0}){\mathrm{e}}^{-\alpha L}}{{(1-{\mathrm{e}}^{-\alpha L/2}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa})}^{2}+4{\mathrm{e}}^{-\alpha L/2}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa}\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\mathrm{\varphi}/2}].$$The reflected intensity is minimum at resonance, i.e., for ${\mathrm{sin}}^{2}\mathrm{\varphi}/2=0$, and maximum at antiresonance, i.e., for ${\mathrm{sin}}^{2}\mathrm{\varphi}/2=1$. The half-width at half-maximum of the resonance corresponds to the value

## (27)

$${\mathrm{\varphi}}_{1/2}=\pi \frac{\mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}},$$## (28)

$${\mathrm{sin}}^{2}\frac{{\mathrm{\varphi}}_{1/2}}{2}=\frac{{(1-{\mathrm{e}}^{-\alpha L}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa})}^{2}}{2[1+{\mathrm{e}}^{-\alpha L}(1-{\gamma}_{0})(1-\kappa )]}.$$## (29)

$$\frac{\mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}\simeq \frac{\sqrt{2}}{\pi}\frac{1-{\mathrm{e}}^{-\alpha L}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa}}{{[1+{\mathrm{e}}^{-\alpha L}(1-{\gamma}_{0})(1-\kappa )]}^{1/2}}.$$## Appendix B:

### Optimization of the Slope of the PDH Error Signal

In this appendix, we derive the slope of the PDH error signal for a single coupler ring cavity such as the one of Fig. 1 and show that the critical coupling is the one that maximizes this slope. We adopt the notations of Ref. 15, where the incident field at angular frequency $\omega =2\pi \nu $, of complex amplitude ${E}_{0}$, is phase modulated at angular frequency $\mathrm{\Omega}$ with an amplitude $\beta $. The incident field thus reads

## (30)

$${E}_{0}{\mathrm{e}}^{i[\omega t+\beta \text{\hspace{0.17em}}\mathrm{sin}(\mathrm{\Omega}t)]}={E}_{0}\sum _{n=-\infty}^{n=\infty}{J}_{n}(\beta ){\mathrm{e}}^{i(\omega +n\mathrm{\Omega})t},$$## (31)

$${E}_{0}[F(\omega ){J}_{0}(\beta ){\mathrm{e}}^{i\omega t}+F(\omega +\mathrm{\Omega}){J}_{1}(\beta ){\mathrm{e}}^{i(\omega +\mathrm{\Omega})t}+F(\omega -\mathrm{\Omega}){J}_{-1}(\beta ){\mathrm{e}}^{i(\omega -\mathrm{\Omega})t}],$$## (32)

$$F(\omega )=\sqrt{1-{\gamma}_{0}}\left[\frac{\sqrt{1-\kappa}-{\mathrm{e}}^{-\alpha L/2}{\mathrm{e}}^{-i\omega /\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}\sqrt{1-{\gamma}_{0}}}{1-{\mathrm{e}}^{-\alpha L/2}{\mathrm{e}}^{-i\omega /\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}\sqrt{1-{\gamma}_{0}}\sqrt{1-\kappa}}\right].$$If we suppose that the sidebands are completely out of resonance, then $F(\omega \pm \mathrm{\Omega})\simeq -1$ and, following Ref. 15, the error signal after demodulation, i.e., multiplication by $\mathrm{sin}(\mathrm{\Omega}t)$, and low-pass filtering is given by

where $G$ is the optical to electrical conversion gain and ${P}_{0}$ is the optical power associated with the incident laser field ${E}_{0}$. The error signal is thus proportional to the imaginary part of $F(\omega )$, which, according to Eq. (32), is given by## (34)

$$\mathrm{Im}[F(\omega )]=\frac{\kappa (1-{\gamma}_{0}){\mathrm{e}}^{-\alpha L/2}\mathrm{sin}(\omega /\mathrm{\Delta}{\nu}_{\mathrm{FSR}})}{{(1-{\mathrm{e}}^{-\alpha L/2}\sqrt{1-\kappa}\sqrt{1-{\gamma}_{0}})}^{2}+4{\mathrm{e}}^{-\alpha L/2}\sqrt{1-\kappa}\sqrt{1-{\gamma}_{0}}{\mathrm{sin}}^{2}(\omega /2\mathrm{\Delta}{\nu}_{\mathrm{FSR}})}.$$If we call $\delta \omega $, the shift of $\omega $ with respect to resonance and suppose that $\delta \omega \ll 2\pi \mathrm{\Delta}{\nu}_{\mathrm{FSR}}$, then Eq. (34) becomes

## (35)

$$\mathrm{Im}[F(\delta \omega )]\simeq \frac{\kappa (1-{\gamma}_{0}){\mathrm{e}}^{-\alpha L/2}}{{(1-{\mathrm{e}}^{-\alpha L/2}\sqrt{1-\kappa}\sqrt{1-{\gamma}_{0}})}^{2}}\frac{\delta \omega}{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}.$$Close to resonance, the evolution of the error signal with the cavity detuning is thus linear. One can optimize the slope of this evolution by chosing the value of the coupling $\kappa $ that satisfies

By taking the derivative of Eq. (35) with respect to $\kappa $, one retrieves exactly the condition (4) for critical coupling:

In the case of critical coupling, Eq. (35) becomes

## (38)

$$\mathrm{Im}[F(\delta \omega )]\simeq \frac{(1-{\gamma}_{0}){\mathrm{e}}^{-\alpha L/2}}{1-{\mathrm{e}}^{-\alpha L}(1-{\gamma}_{0})}\frac{\delta \omega}{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}.$$In the case of a high finesse cavity, we have ${\mathrm{e}}^{-\alpha L}(1-{\gamma}_{0})\simeq 1-\alpha L-{\gamma}_{0}$, so that

## (39)

$$\mathrm{Im}[F(\delta \omega )]\simeq \frac{1}{\alpha L+{\gamma}_{0}}\frac{\delta \omega}{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}.$$Besides, Eq. (5) leads to

## (40)

$$\frac{1}{\mathcal{F}}=\frac{\mathrm{\Delta}{\nu}_{\mathrm{FSR}}}{\mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}\simeq \frac{\alpha L+{\gamma}_{0}}{\pi},$$## (41)

$$\mathrm{Im}[F(\delta \omega )]\simeq \frac{\delta \omega}{\pi \mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}.$$Using Eq. (33), we finally obtain:

## Appendix C:

### Shot Noise Limit

In this appendix, we use the slope error signal expression obtained in Appendix B to derive the shot noise limit of a resonant gyroscope. The detection of the power ${P}_{\mathrm{det}}$ reflected by the cavity creates a photocurrent given by

where $\chi $ is the detector efficiency and $\lambda $ is the light wavelength. The shot noise associated with this current is given by where $\tau $ is the integration time, leading to the following noise equivalent power:## (45)

$${P}_{\mathrm{SNL}}=\frac{{i}_{\mathrm{SNL}}}{\chi e\lambda /hc}=\sqrt{\frac{2hc}{\chi \lambda \tau}{P}_{\mathrm{det}}}.$$The noise on the servoloop error signal is thus

## (46)

$${\u03f5}_{\mathrm{SNL}}=\frac{G{P}_{\mathrm{SNL}}}{\sqrt{2}}=G\sqrt{\frac{hc}{\chi \lambda \tau}{P}_{\mathrm{det}}},$$## (47)

$$\delta {\omega}_{\mathrm{SNL}}=\sqrt{\frac{hc}{\lambda \chi \tau}}\sqrt{{P}_{\mathrm{det}}}\frac{\pi \mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}{2{P}_{0}|{J}_{0}(\beta ){J}_{1}(\beta )|}.$$The Sagnac effect scale factor relates a rotation rate $\dot{\theta}$ into a frequency difference $\mathrm{\Delta}{\omega}_{\mathrm{Sagnac}}$ through

Then, Eq. (46) translates into the following noise limit on the measurement of a variation of the rotation rate:

## (49)

$$\delta {\dot{\theta}}_{\mathrm{SNL}}=\frac{{n}_{0}\lambda}{2\pi D}\sqrt{2}\delta {\omega}_{\mathrm{SNL}}\text{\hspace{0.17em}},$$## (50)

$$\delta {\dot{\theta}}_{\mathrm{SNL}}=\frac{{n}_{0}\mathrm{\Delta}{\nu}_{\mathrm{FWHM}}}{D}\sqrt{\frac{\lambda hc}{\chi \tau}}\frac{\sqrt{{P}_{\mathrm{det}}}}{2\sqrt{2}{P}_{0}|{J}_{0}(\beta ){J}_{1}(\beta )|}.$$Close to resonance, in critical coupling conditions, the power falling on the detector is mainly due to the two first sidebands, which are fully reflected by the cavity, leading to

Moreover, as shown in Ref. 15, the error signal can be maximized by choosing $\beta \simeq 1.08\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}$, for which ${J}_{0}{(\beta )}^{2}\simeq {J}_{1}(\beta )\simeq 1/2$. In these conditions, Eq. (50) becomes

which is equivalent to Eq. (11) above.## Appendix D:

### Kerr Bias

As shown in Ref. 2, if the two counter propagating intracavity beams exhibit a power difference $\mathrm{\Delta}P$, the Kerr effect creates a difference between the refractive indices seen by the two counter propagating waves that reads

where ${n}_{2}$ is the waveguide effective nonlinear index and $\sigma $ is the guided mode area. This leads to a difference between the resonance frequencies of the two counterpropagating modes, given by## (54)

$$\mathrm{\Delta}{\omega}_{\mathrm{Kerr}}=2\pi \frac{c{n}_{2}}{\lambda {n}_{0}}\frac{\mathrm{\Delta}P}{\sigma}.$$Using the gyro scale factor given by Eq. (48), one obtains the expression of the bias angular velocity induced by the Kerr effect:

which can be related to the difference $\mathrm{\Delta}{P}_{0}$ between the powers incident on the cavity in the CW and CCW directions using Eqs. (8) and (10), leading to## (56)

$${\dot{\theta}}_{\mathrm{Kerr}}=\frac{c{n}_{2}\u200a\mathcal{F}}{\pi \sigma D}\frac{\mathrm{\Delta}{P}_{0}}{2}.$$The extra factor of 2 at the denominator of Eq. (56) is due to the factor ${J}_{0}{(\beta )}^{2}$, which relates the carrier power to the total incident power ${P}_{0}$ in the PDH servolocking configuration [see Eq. (31)] with $\beta =1.08\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{rad}$.

## Appendix E:

### Calculation of Brillouin Scattering Threshold

The threshold pump power for Brillouin scattering in a single-pass geometry in a waveguide of length $L$ is approximated in Ref. 55 by the formula

where $\sigma $ is the mode area, ${L}_{\mathrm{eff}}=(1-{\mathrm{e}}^{-\alpha L})/\alpha $ is the waveguide effective length taking propagation losses into account, and $g$ is the Brillouin gain ($g=5\times {10}^{-11}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{W}$ for silica). With the SiN waveguides considered in Sec. 4, we have $\sigma =33\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$. Since $\alpha =0.0736\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{-1}$ for $0.32\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dB}/\mathrm{km}$ losses (see Table 1) and since the cavity diameter $D$ is only few cm large, we have ${L}_{\mathrm{eff}}\simeq \pi D$. Assuming, for instance, a 5-cm diameter cavity and that the low confinement of the mode allows us to consider that it propagates in silica only, we end up with a threshold of 100 W, far above the considered powers. For this calculation, we considered a single-pass geometry because we can choose the cavity free-spectral range to make the Brillouin scattered frequency nonresonant when the laser beam is resonant, as opposed to a doubly resonant geometry.^{56}

## Acknowledgments

We wish to thank Jérôme Bourderionnet, Alfredo De Rossi, and Sylvain Combrié for sharing their expertise on integrated photonics technologies with us. This work is supported by the Agence Nationale de la Recherche (Project PHOBAG: ANR-13-BS03-0007) and the European Space Agency (ESA). The work of G.F., A.R., and F.B. is performed in the framework of the joint research lab between Thales Research & Technology and Laboratoire Aimé Cotton.

## References

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## Biography

**Gilles Feugnet** graduated from Institut d’Optique Graduate School in 1989. He then joined Coherent Inc. (working on the development of mode-locked titanium sapphire laser), and in 1991, he joined Thales Research & Technology (formely Laboratoire Central de Recherches). He has been working on laser-diode pumped solid-state lasers such as for gyrolasers. He is the author or coauthor of more than 40 papers and holds 20 patents.

**Alexia Ravaille** graduated from Institut d’Optique Graduate School in 2015. She is currently preparing her PhD degree on resonant optical passive gyroscope at Laboratoire Aimé Cotton, Orsay, in close collaboration with Thales Research & Technology.

**Sylvain Schwartz** graduated from Ecole Polytechnique in 2001 and received his PhD degree from the same institution in 2006. He joined Thales Research and Technology (Palaiseau, France) in 2006 as a research engineer. He is the author or coauthor of about 15 journal papers and holds 25 patent applications. Since 2015, he has been working in the group of Mikhail Lukin at Harvard. His current research interests include optical gyroscopes, atom interferometry for inertial sensing, and quantum information with neutral atoms coupled through Rydberg interaction and photonic crystal cavities.

**Fabien Bretenaker** graduated from Ecole Polytechnique, France, in 1988. He received his PhD from University of Rennes, Rennes, France, in 1992 while working on ring laser gyroscopes for Sagem. He joined the Centre National de la Recherche Scientifique, Rennes, in 1994 and worked in Rennes until 2002 on laser physics and nonlinear optics. In 2003, he joined the Laboratoire Aimé Cotton, Orsay, France, working on nonlinear optics, laser physics, quantum optics, and microwave photonics. He is also a part time professor in Ecole Polytechnique, Palaiseau, France, and an adjunct professor in Ecole Normale Supérieure Paris Saclay, Cachan, France.