at high frequencies; it also reduces the amplitude fluctuations at low frequencies as its modulus is less than 0 dB. $H_1$ is related to cavity parameters, to the acoustic damping rate, and to the coupling ratio. $H_2$ corresponds to the delay line filtering effect of the interferometer (fiber spool ring cavity). The phase noise reduction between the pump and Stokes is called the K factor and is expressed as :

$$K = \left( 1 - \frac{\beta_A}{\ln(R)} \right)^2 \qquad (3.5)$$

As cavity losses and gain bandwidth are taken into account (table (3.2)), frequency noise reduction of 37 ± 1 dB with such a Brillouin laser is expected. A similar frequency noise reduction effect was reported in [33] and [34]. The reduction factor may vary with the feedback rate R.

In Brillouin laser, there are three main sources of noise. The first is the noise from the pump laser; this noise will be filtered out before being transferred to the Stokes wave as mentioned in the former subsection. To get a low-noise Brillouin laser, the ideal configuration would be to use a low-noise pump laser. The second source is related to the cavity; it is mainly a variation of cavity length and reinjection rate due to thermal and mechanical fluctuations. If the laser cavity is sufficiently stable, then the fluctuations of fiber length are small, which induces only "slow" drift of the laser frequency, typically in the frequency band (DC-1 kHz). On the other hand, they do not contribute to the intrinsic linewidth of the Stokes wave. The third source is related to the spontaneous Brillouin scattering. By analogy with conventional lasers, for which spontaneous emission limits the Schawlow-Townes linewidth, similarly spontaneous Brillouin scattering limits the Stokes-wave coherence. The noise level is generally very low when compared to the other noise contributions.

In the following, we study the frequency noise of the non-resonant Brillouin laser in three different configurations. The first one consider a free running laser to pump the non-resonant BFL cavity. In the second case, a stabilized pump laser is used and then, in the third configuration we study a laser architecture based on the cascading of two non-resonant BFL. We conclude by a comparison of the frequency noise reduction performances. We measure the frequency noise via a self-heterodyne technique thanks to a Mach-Zehnder interferometer with a 200 meter delay, which was introduced in section (1.4.3).
### 3.6.2 Non-resonant BLF frequency noise using a free running pump laser

Tunics OM laser is used as a pump laser for this experiment. Measured frequency noise of Stokes and pump laser are plotted on figure (3.9a) (in black for the pump and blue for $S_1$). As we have seen before the BFL frequency noise is reduced by a factor of K due to the combined influences of the cavity feedback and acoustic damping rate. In order to compare the experimental results, we analytically calculated the frequency noise by the product of the pump frequency noise and the analytical transfer function of the BFL (dotted blue line). A good match is found between measured and the calculated frequency noise. A reduction of 37 dB relatively to the pump frequency noise over a frequency range comprise between 1 kHz and 300 kHz is observed. Below 1 kHz, we remark a discrepancy between the measurement and the simulation. Indeed, the amplitude of the measured frequency noise is slightly larger because measurements are limited by noise floor. In figure (3.9), dotted gray line shows the noise floor of MZI with 200 m fiber (see section (1.4.4b)). Below 8 kHz, noise measurements are limited by the noise floor of our experimental bench. This is due to the stability of our MZI. Note that a better stabilization of the interferometer could improve the noise floor [171]. Above 100 kHz, the noise floor is nearly 24 dB down from the frequency noise of Stokes.

<figure><img src="image_45.png" /><figcaption>Figure 3.9: Experimental and analytical frequency noise measurement of 20 m non-resonant BFL. Corresponding noise floor with 200 m delay is plotted in dotted line.</figcaption></figure>

We estimate the integrated linewidth of the pump laser and of the Stokes line using the "*β-line*" approximation. In the free-running (FR) operation, the pump laser exhibits an integrated linewidth of 187 ± 18 kHz for 10 ms observation time. The frequency noise
level is $3 \times 10^6 \text{ Hz}^2/\text{Hz}$ at 100 Hz offset frequency. From the estimated frequency noise reduction previously calculated, we may expect to observe an integrated linewidth of 360 Hz (see blue dotted line in figure (3.9)). Taking into account the noise floor, that gives frequency noise level of $5 \times 10^2 \text{ Hz}^2/\text{Hz}$ at 100 Hz offset frequency, enables us to say that the integrated linewidth is below 2 kHz and that we cannot know its exact value. Thus it has decreased the integrated linewidth from $187 \pm 18 \text{ kHz}$ to a value below 2 kHz. If we assume that it is correct to draw a comparison between the analytical results and the noise floor, the noise floor has to be improved by 20 dB in order to measure the low frequency component (and the following case shown in figure (3.10a) by 40 dB).

### 3.6.3 Non-resonant BFL frequency noise using stabilized pump laser

To suppress mode-hopping, we stabilize the pump frequency onto a cavity-resonance frequency. Figure (3.10) shows the frequency noise measured for the free running pump laser, frequency-locked (FL) pump, and the SBS laser. By locking the pump laser onto a cavity resonance, the integrated linewidth of the pump laser is improved, when comparing the black curve and red curve of figure (3.10). Once again the improvement below 8 kHz cannot be characterized because the PSD reaches the noise floor. We can only say that the integrated linewidth is lower than $66.17 \pm 6 \text{ kHz}$ kHz. The improvement of the pump noise level is at least the one of the noise floor, which is decreased at least by three order of magnitude compared to frequency noise of free-running pump.

As we have seen previously, SBS laser will improve the reduction of the laser linewidth due to the improvement of pump flicker noise. Expected frequency noise are plotted in lightgreen colour. Unfortunately, $S_1$ (FL) is limited by noise floor of our bench. So we do not see any improvement in the $S_1$ frequency noise between FR and FL. Still, the intrinsic linewidth is reduced from 6.8 kHz to 1.3 Hz, which corresponds to 37 dB reduction. It matches our estimated value of $37 \pm 1$ dB.
<figure><img src="image_46.png" /><figcaption>Figure 3.10: Frequency noise of the non-resonant BFL laser. (a) pump and Stokes signal with and without stabilization in logarithmic scale (b) frequency noise of cascaded non-resonant BFL laser in linear scale.</figcaption></figure>

### 3.6.4 Frequency noise performances of two cascaded non-resonant BFL

To find the intrinsic linewidth limitations, we cascaded two non-resonant BFL, frequency noise is plotted in figure (3.10b). A first Stokes wave is generated with a 20 meters fiber cavity, which gives 37 dB noise reduction. Then the output of this BFL is amplified using an EDFA and injected into a second BFL. The second Brillouin laser is made by a 14 meter fiber cavity and 75 % of rejection rate, which corresponds to a noise reduction factor of 33 dB. Then a reduction factor close to 70 dB might be expected when compared to the pump laser frequency noise.

We plot on figure (3.10b) the expected noise reduction (dotted brown curve) and the measured Stokes frequency noise at the output of the two cascaded Brillouin laser (cyan curve). We obtained 61 dB reduction in frequency noise. This performance is limited by the noise floor of our MZI (dotted gray curve) that do not allows to completely characterize this configuration. The cascaded BFL reaches the intrinsic linewidth of 47 mHz at 100 kHz.

We have shown that cascaded Brillouin laser architecture laser can reach the sub-Hz intrinsic linewidth. However we did not succeed to explore the limitation of this Brillouin laser configuration.
### 3.7 Conclusion

Non-resonant Brillouin laser is demonstrated using a silica fiber cavity. We explained the threshold power and mode-hopping characterization of a Brillouin ring-cavity. 20 meters Brillouin cavity is made using a silica fiber and it is experimentally shown that even a short cavity could suffer from mode-hopping due to the thermal shift of the cavity resonances. We then proposed a frequency stabilized non-resonant Brillouin laser design, which suppressed the mode-hopping effect. Also, it improved the pump laser noise by transferring the stability of the cavity to the pump using the PDH method. Finally, the frequency noise properties of a non-resonant Brillouin fiber laser have been investigated experimentally with the realization of Hz level Brillouin laser. Moreover, we demonstrated that a mHz linewidth laser based on cascading of two BFL cavities can be easily reached.
<figure><img src="image_47.png" /></figure>
# Chapter 4

## Multi-Stokes Brillouin fiber laser

### 4.1 Introduction

We have seen that stimulated Brillouin scattering can be used as a laser gain medium and that non-resonant Brillouin laser can exhibit a sub-Hz linewidth. The high laser threshold of the non-resonant Brillouin laser can be a limiting factor in some applications. Another class of high performances lasers are resonant Brillouin lasers, in which pump and Stokes waves are resonant inside the cavity. Resonant Brillouin laser acquire a lot of interest in laser physic communities for its ability to exhibit low threshold and low noise dynamics [25, 123]. Resonant Brillouin laser is capable to produce different lasing Stokes orders. Power dynamics, linewidth narrowing and intensity noise reduction have been demonstrated in various systems as fiber rings [34, 126], whispering gallery mode microresonators [158] or planar waveguide structures [172] to only name a few. Those systems are made of nonlinear materials as silica [34], chalcogenide [33], silicon nitride [163], crystal fluoride [145] or silicon [173].

One remaining question concerning the resonant Brillouin laser architecture is the potential to improve the intensity and frequency noise reduction by cascading the SBS process. Theoretical papers related to this problematic predicts intensity and frequency noise dynamics in the cascaded SBS process [3]. In this chapter, we demonstrate experimentally, that under a specific operating regime, Stokes line intensity noise can be reduced up to 20 dB with respect to the input pump RIN. Moreover we report the complex RIN dynamics of a multi-Stokes Brillouin laser (MBL) in various configurations. Those results have been published in "Optics Express" in December 2018 [174]. Similarly, we investigate
the frequency noise dynamics of the multi-Stokes Brillouin laser. We observe a close to 27 dB of frequency noise reduction for single mode Brillouin laser compare to it's pump noise. In particular, we investigate the roles of the photon lifetime and detuning of the gain bandwidth on the RIN and frequency noise reduction of the MBL.

The chapter is organized as follows. In section (4.2), we describe the physics of the MBL. In section (4.3), the experimental setup and the fiber ring resonator used for the present study are explained. We introduce the MBL model based on coupled mode equations and derive the stationary response of MBL, which are explained in sections (4.4) and (4.5) respectively. In section (4.6), we study the RIN behavior of Stokes lines in function of the input pump power in presence of higher order Stokes lines. Moreover, we discuss the impact on RIN performances of several parameters as cavity lifetimes, detuning of gain bandwidth and pump noise level.

In order to study the influence of detuning and length, we setup a short cavity with high-Q factor, which enables the detuning to be controlled by pump wavelength. In section (4.7), we introduce the frequency noise study of the MBL. First, we describe the frequency noise behavior of the Stokes lines as a function of the input pump power. Secondly, we explain the over all frequency noise behavior of individual Stokes lines and the impact of the cascaded and gain detuning effects. Finally, we study the impact of photon lifetime of the cavity on frequency noise of the Stokes 1 (S1) laser emission.

## 4.2 Resonant Brillouin laser

In resonant Brillouin laser configuration, pump and Stokes waves are resonant inside the cavity. The critical physics behind the Brillouin laser is the coupling between optical and acoustic waves. As we have seen in chapter 2, through Brillouin scattering process, a high-frequency photon (Pump) characterized by its frequency $\omega_0$ and wavevector $k_0$, decays to a lower frequency photon (Stokes) and phonon with frequencies $\omega_1$ and $\Omega_0$ and wavevectors $k_1, q_0$, respectively. Brillouin coupling efficiency transfers the energy from the pump to the Stokes waves. Stokes photons can stimulate as well the decay of pump photons to Stokes photons generation, similarly to a classical gain media through stimulated optical amplification. This amplification occurs in a small bandwidth $\Delta\nu_B$, which is related to the phonon decay rate in the medium. When the medium is placed inside a cavity, the process will be more efficient if the pump is coupled to an optical resonant mode. The
<figure><img src="image_48.png" /><figcaption>Figure 4.1: Illustration of cascade Brillouin laser. A pump laser of frequency $\omega_0$ is injected into an optical resonator. When its wavelength matches the Brillouin condition (($\Omega_B/2\pi$) = $N \times \Delta\nu_{FSR}$), pump resonant mode at $\omega_0$ is scattered at $\omega_1$ by emitting a phonon at $\Omega_B$. When the pump power is sufficiently high, the Stokes wave at $\omega_1$ acts as a pump for higher order Stokes lines.</figcaption></figure>

Stokes signal is generated at a frequency related to one of the resonant modes. If the Stokes frequency is at the maximum gain frequency, then the Brillouin frequency shift is a multiple of the free spectral range (($\Omega_B/2\pi$) = $N \times \Delta\nu_{FSR}$) as shown in figure (4.1). Then the Stokes wave generation is more efficient. At higher pumping power, counter-propagating Stokes 1 (S1) acts as a pump for a counter-propagating second Stokes 2 (S2). This process is generated by a distinct phonons reservoir with frequency $\Omega_1$ and is propagating in the opposite direction to phonons at frequency $\Omega_0$. Frequency differences between the consecutive Stokes are not equal and are slightly shifted because each Stokes wave is created with different phonons frequencies [34]. The shift is moreover dependent of the pump wavelength and temperature as we have seen in section (3.2).

The optical pump is thus injected into a high-Q cavity. As underlined before, the Brillouin gain, created by the pump wave circulating several times in the cavity, allows a Stokes wave flowing in the opposite direction to reach the laser threshold. We consider that the pump frequency and cavity resonance frequency are stable. Resonant Stokes 1 (S1), located at the maximum of the Brillouin gain, can then be amplified. It propagates in the opposite direction to that of the pump. Similarly, when the power in S1 reaches the output power needed for Stokes 2 (S2), to compensate its linear losses, coherent emission of S2 wave also takes place; the cascaded process is on.
## 4.3 Brillouin laser architecture

We have seen that different types of Brillouin laser architectures have been studied in the last few years. In this thesis, we study two kind of Brillouin laser, resonant and non-resonant configuration. In the previous chapter, we have discussed and studied the dynamics of the non-resonant Brillouin laser. In this section, we examine the resonant Brillouin laser dynamics, which is interesting due to the low threshold and higher order Stokes emissions. We are focusing on the fiber ring cavity based Brillouin laser. This section is organized in three parts. Firstly, we will explain the fiber ring cavity and its parameters, which are essential part of the Brillouin laser. Secondly, we will describe the experimental setup of the laser. Finally, we will detail the laser locking method used to maintain the pump laser in resonance with the fiber ring cavity.

### 4.3.1 Fiber Ring Cavity

The central element of the MBL is composed of a fiber ring resonator. As we have seen in previous chapters, the gain or losses inside the resonator is quantified by the intrinsic cavity lifetime ($\tau_0/2$) and the coupling strength by the coupling lifetime ($\tau_e/2$). Moreover, the resulting total photon lifetime of the resonator can be linked to the Q-factor (see section (2.4.1)).

The resonator is made of a 20 meters polarization-maintaining fiber corresponding to a free-spectral range (FSR) of 10 MHz. The light is coupled in and out through a variable coupler allowing to vary the coupling coefficient and then the Q-factor of the resonator [97]. In the present study, we consider two resonator configurations for study the impact of photon lifetime in Brillouin lasers, which we call: the high-Q resonator ($Q_h = 6 \times 10^8$) and the low-Q resonator ($Q_l = 2 \times 10^8$). Table (4.1) shows the extracted cavity parameters using the cavity ring down method, which was introduced in section (2.4). The low-Q factor ($MBL_{LQ}$) and high-Q factor ($MBL_{HQ}$) cavities are in the overcoupling regime ($\tau_0 > \tau_{el}$) and the undercoupling regime ($\tau_0 < \tau_{eh}$) respectively.

In terms of Brillouin laser performances, the overcoupling regime implies a higher output power at the price of higher lasing threshold with respect to the undercoupling operation. As it will be shown later, the Q-factor of the resonator impacts the RIN features of the MBL (Section 4.6).
<table><caption>Table 4.1: Parameters of resonant BFL 20m cavity</caption><thead><tr><th>Parameters</th><th>MBL<sub>HQ</sub></th><th>MBL<sub>LQ</sub></th></tr></thead><tbody><tr><td>τ<sub>0</sub></td><td>1.39 µs</td><td>1.39 µs</td></tr><tr><td>τ<sub>e</sub></td><td>3.7 µs</td><td>0.47 µs</td></tr><tr><td>Q-factor</td><td>6 × 10<sup>8</sup></td><td>2 × 10<sup>8</sup></td></tr></tbody></table>

<figure><img src="image_49.png" /><figcaption>Figure 4.2: Experimental setup to study the lasing properties of multi-Stokes Brillouin Laser. Pump: Koheras continuous wave laser, EDFA: Erbium-doped fiber amplifier, VOA-variable optical attenuator, VC: Variable coupler, PZT: Piezoelectric transducer, Filter: Yenista optical filter, PD: Photodetector, PID: Proportional-integral differential amplifier, HV: High-voltage amplifier, TA: Transimpedance Amplifier, ESA: Electrical spectrum analyzer. 20 m polarization maintaining fiber spooled in PZT. Red line: Pump wave, Blue: Stokes 1 wave, Green: Stokes 2 wave.</figcaption></figure>

### 4.3.2 Experimental setup

The experimental setup is illustrated in figure (4.2). The fiber ring cavity is optically pumped using a continuous wave (CW), that has an integrated linewidth of 2 kHz with observation time 10 ms at 1552.4 nm. The laser output power is amplified before the cavity injection. A motorized variable optical attenuator helps to control the input power in the MBL. 1 % of the input pump power is collected for power monitoring.

The laser pump signal is injected in the cavity via a circulator and a coupler, and then propagates clockwise around the cavity (red arrow). The circulating pump wave initiates Brillouin amplification in S1 wave, traveling in the opposite direction (blue arrow). For
sufficiently high pump power ($P_{in} = P_{th}$), Brillouin lasing threshold is reached giving rise to efficient S1 emission outside the resonator. S1 wave can, itself, play the role of pump laser for S2 wave (green arrow).

### 4.3.3 Cavity servoing

The laser threshold for the various Stokes waves can be drastically reduced when the pump signal is resonantly coupled to the cavity. This configuration can be achieved by locking a cavity resonance to the CW pump laser. We implement a derivative spectroscopy stabilization method [175]; its setup is presented in gray box in figure (4.2c). Locking technique was explained in section (1.5.1). The actuator consists in a piezoelectric (PZT) ceramic cylinder wrapped by few meters of fiber. Varying the voltage on the piezoelectric cell implies a modification of the cavity length and then a control of the frequency position of the cavity resonances. We apply a slow modulation of 10 kHz on the PZT through a high-voltage amplifier, which creates a derivative error signal from the lock-in amplifier. Actuator signal is produced by sending the error signal to a PID controller. Using this technique, the cavity is maintained in resonance with the pump laser during MBL operation.

As can be seen in figure (4.2a), from the observation at the Brillouin optical spectrum analyzer (BOSA), a multi-Stokes spectrum contains higher-order Stokes waves when the pump power injected approaches 200 mW in a $MBL_{HQ}$ cavity. The first-order Brillouin offset is about 10.87 GHz. In conclusion, we introduce the Brillouin laser architecture and cavity parameters, which will be used for the analysis of the power and RIN dynamics of the laser. In the next section, we will introduce the theoretical model for the analysis of resonant Brillouin laser dynamics.

## 4.4 Theoretical model

Cascaded Brillouin models are reported in the literature [3, 33, 34, 116]. To study the dynamics of the cascade Brillouin process, we used the coupled-mode formalism to describe the SBS interaction between the pump, Stokes and density waves for a ring resonator. W. Loh et al. [4] described single Stokes generation of Brillouin laser model with the help of a three wave equations. Irina Balakireva (post-doc in our lab) extended the model
into a multi-Stokes Brillouin laser (MBL) configuration [174]. The model, based on the coupled modes formalism [176], depicts the temporal dynamics of 2N + 1 coupled-mode equations where N is the maximum number of Stokes waves under consideration:

$$ \frac{\partial A_0}{\partial t} = -\frac{1}{\tau}A_0 - iq_0\omega_0 A_1\rho_1 + \sqrt{\frac{2}{\tau_e}}Se^{i\sigma_0 t}, \quad (4.1) $$

$$ \frac{\partial A_{\eta}}{\partial t} = -\frac{1}{\tau_{\eta}}A_{\eta} - iq_{\eta}\omega_{\eta}[A_{\eta-1}\rho_{\eta}^{*} + \delta_{\eta \neq N}A_{\eta+1}\rho_{\eta+1}], \quad (4.2) $$

$$ \frac{\partial \rho_{\eta}}{\partial t} = i \frac{\Omega_B^2 - \Omega_{\eta}^2}{2\Omega_{\eta}} \rho_{\eta} - \frac{\Gamma_B}{2} \rho_{\eta} - ip_{\eta} A_{\eta-1} A_{\eta}^*. \quad (4.3) $$

Here $A_\eta$, $\omega_\eta$, and $\tau_\eta$ are called the amplitude, frequency and the lifetime of the generated Stokes wave label $\eta$ (with $\eta \in [1,N]$). $|A_\eta|^2$ is proportional to an energy. Eq. (4.1) describes the dynamics of the pumped mode $A_0$, with an angular frequency emitting in the C band fixed to $\omega_0 = 1.22 \times 10^{15}$ rad/s. $\rho_\eta$ and $\Omega_\eta$ are the amplitude and frequency of the density wave labeled $\eta$. $\Gamma_B$ and $\Omega_B$ are the angular loss rate and frequency related to Brillouin gain bandwidth and frequency shift described before. These two parameters are experimentally extracted (see section 2.3.1). For sake of simplicity, we will consider a constant Brillouin Stokes shift over the cascaded process, which means, $\Omega_\eta = \Omega_B$, then the $\omega_\eta = \omega_{\eta-1} + \Omega_B$ for all $\eta$. Coefficients $q_\eta$ and $p_\eta$ are:

$$ q_{\eta} = \frac{\gamma_{e}}{4n_{0}^{2}\rho_{0}}\Lambda_{\eta}, \quad \text{and} \quad p_{\eta} = \frac{\epsilon_{0}\gamma_{e}n_{0}^{2}\Omega_{\eta}}{4v^{2}}\Lambda_{\rho_{\eta}}, \quad (4.4) $$

where $\gamma_e = 1.5$ is the electrostrictive constant, $\epsilon_0$ is the vacuum permittivity, $n_0$ is the refractive index for silica, $\rho_0$ is the equilibrium density of the material, $v$ is the velocity of the acoustic wave.

In the present work, the resonator is composed of fibers. We can reasonably consider that modes are similarly confined in the fiber and we can then fix the value of mode overlaps $\Lambda_\eta$ and $\Lambda_{\rho_\eta}$ to 1. From this assumption, equations (4.4) will be simplified as $q_\eta$ is constant and equal to $q$ for all waves as it is for $p_\eta = p$ as we assume an equal Brillouin shift for all orders. The amplitude of the pump wave is $S = \sqrt{P_{in}/(V_{ph}\epsilon_0)}$ ($W/m^2$), where $P_{in}$ is the power of the laser (in Watts) and $V_{ph} = 1.56 \times 10^{-9} m^3$ is the optical mode volume of the fiber ring cavity. $P_S$ is the optical power for the Stokes. We will introduce in the following $P_C$ as the clamping power.

As we have seen in chapter 2 (section 2.4), the total lifetime $\tau_\eta$ of each Stokes waves can be found by $1/\tau_\eta = 1/\tau_{0\eta} + 1/\tau_{e\eta}$ where $\tau_{0\eta}$ is the intrinsic lifetime (cavity losses) and $\tau_{e\eta}$ is the coupling lifetime. We consider the lifetimes equal for all waves, and we
<figure><img src="image_50.png" /><figcaption>Figure 4.3: Steady-state laser powers for the Stokes components, obtained from the simulation.</figcaption></figure>

write $\tau_{\eta} = \tau$, $\tau_{0\eta} = \tau_0$, $\tau_{e\eta} = \tau_e$. These last equations have to be linked to the former assumption concerning the confinement. As far as we consider similar confinement, $p$ and $q$ are the same for all waves.

The stationary response can be found by setting all derivatives in Eqs. (4.2) & (4.3) to zero and by finding the joint solutions of the equations. The stationary solutions can be found for any $N$ using three equations. Two of them do not depend on the parity of $N$:

$$|A_{N-1}|^2 = \frac{K_N}{\omega_N}, \qquad (4.5)$$

$$|A_{\eta+2}|^2 = |A_{\eta}|^2 - \frac{K_{\eta+1}}{\omega_{\eta+1}}. \qquad (4.6)$$

Another one is changing following the parity of maximum Stokes order $N$:

$$|A_0|^2 = \frac{2\tau^2}{\tau_e} |S|^2 \times [K^{-1}\omega_0|A_1|^2 + 1]^{-2}, \quad \text{if } N = 2,4,6\dots \qquad (4.7)$$

$$|A_1|^2 = \left[ \frac{\tau}{\sqrt{\tau_e/2} |A_0|} - 1 \right] \frac{K}{\omega_0}, \quad \text{if } N = 1,3,5\dots \qquad (4.8)$$

here $K_{\eta}$ equals to $\Gamma_B/(2\tau p_{\eta}q)$. The technique to calculate the amplitude of the laser emission is the following. First, we calculate $|A_{N-1}|^2$ using Eq. (4.5), and after using Eq. (4.6) we find $|A_{N-3}|^2, |A_{N-5}|^2$ etc. until $A_0$ if $N$ is odd or $A_1$ if $N$ is even. After we use Eqs. 4.7 or 4.8 and finish the calculations of missing lines amplitude by Eq. (4.6) until $|A_N|^2$.

We plot in figure (4.3), the evolution of various Stokes order versus a S1-threshold-normalized input pump power. For a normalized pump power $P_{pump} = P_{in}/P_{th}$ lower
than 1, the transmitted pump power evolved linearly. At $P_{pump} = 1$, the transmitted pump is clamped. This operating point corresponds to the onset of coherency in the first Stokes order. Indeed, the Brillouin gain, induced by the pump, balances the linear losses experience by the mode $A_1$. An efficient stimulated Brillouin emission occurs (blue line). When the power in S1 reaches the normalized output power needed to compensate linear losses of S2, coherent emission also takes place for S2 (green line), then the cascaded process is activated.

The cascaded generation of multiple Stokes waves has been treated analytically by Toyama et al. [116] considering resonant pumping of the cavity and constant Brillouin gain for all Stokes orders. The model gives rise to similar results than previously reported models in the literature [3, 116]. We will see in the following that this analytical model gives good matches with the experimental results.

## 4.5 Power characterization of multi-Stokes Brillouin fiber laser

This section is dedicated to the experimental analysis of the power dynamics of the MBL and its comparison with analytical results presented in the above section. In Fresnel's thesis [34], the power characterization of 4 Stokes orders and a study of the efficiency and threshold have been reported. A similar power dynamics is reported in microsonator [147] and integrated ring Brillouin laser [163]. These recent studies give quantitative picture of the power dynamics of the Brillouin laser. In this section, we explicitly detail power dynamics of the MBL as it is of fundamental importance for MBL noise analysis, which are closely interconnected. Moreover, we will compare experimental and theoretical results. Also we will present the influence of the photon lifetime on the threshold and output power of Stokes signals.

The experimental setup of MBL has been detailed in section (4.3); for power characterization study, we locked pump-laser frequency on $MBL_{LQ}$ cavity by a top-of-fringe method. The pump power is varied from zero to 200 mW, Stokes signal is filtered and measured using a power meter. The evolution of output Stokes lines 1, 2, and 3 as a function of the input pump power is given for the low-Q factor cavity in figure (4.4a). We obtain 26.5 mW of threshold for S1 signal. The S1 signal linearly increases above this
<figure><img src="image_51.png" /><figcaption>Figure 4.4: (a) Power characterization of *MBL*<sub>LQ</sub> cavity (b) Normalized Power characterization of *MBL*<sub>LQ</sub> cavity with its input power, *P*<sub>c</sub>: clamping power, *P*<sub>th</sub>: threshold power of S1</figcaption></figure>

threshold for increasing pump power. At 4 times the threshold corresponding to 105 mW, the S2 linearly increases with input pump because it overcomes the linear cavity losses. At the same pump power, S1 experiences a clamping effect. That maintains constant the S1 output power until the onset of coherency for S3 line for a normalized pump power equals to 212 mW (8 Pth). Similarly, the cascaded process takes place. In figure (4.4b), a comparison is drawn between experiment and theory for *MBL*<sub>LQ</sub>. Straight lines and stars correspond to experimental results and numerical simulations, respectively. The pump power *P*<sub>in</sub> is normalized to the first Stokes power threshold. Also the Stokes output power (*P*<sub>s</sub>) is normalized with the clamping power (*P*<sub>c</sub>). When the power in S1 reaches the normalized output power needed to compensate S2 linear losses, coherent emission (green line) also takes place.

<figure><img src="image_52.png" /><figcaption>Figure 4.5: Power characterization of *MBL*<sub>HQ</sub> cavity</figcaption></figure>
It is worth to mention that when the output power of a Stokes order ($N + 1$) evolves monotonically with the pump power, the output power of the pumping Stokes order ($N$) is constant due to the clamping effect [177]. Stokes lines output power alternates between monotonous evolution and continuous power emission when the gain clamping occurs due to the onset of coherent radiation on the $N + 1$ Stokes wave.

In order to study the influence of the photon lifetime, we have characterized the High-Q cavity ($MBL_{HQ}$) and shown the Stokes output power as a function of the input pump power in figure (4.5), (lines 1, 2, 3 and 4). The first Stokes (S1) laser threshold reduced from 26.5 mW to 7.2 mW by using a high-Q resonator ($Q_h = 6 \times 10^8$) instead of a low-Q resonator ($Q_l = 2 \times 10^8$). Stokes thresholds are summarized in the table (4.2). Similarly, slope efficiency of S1 is reduced from 37% (low-Q) to 13% (high-Q) because the output coupling power is decreased.

<table><caption>Table 4.2: Threshold of resonant BFL cavity</caption><thead><tr><th>order</th><th>P<sub>in</sub>/P<sub>th</sub></th><th>MBL<sub>HQ</sub>(mW)</th><th>MBL<sub>LQ</sub>(mW)</th></tr></thead><tbody><tr><td>S1</td><td>1</td><td>7.2</td><td>26.2</td></tr><tr><td>S2</td><td>4</td><td>28.7</td><td>104.8</td></tr><tr><td>S3</td><td>8</td><td>57.2</td><td>209.4</td></tr><tr><td>S4</td><td>18</td><td>130.1</td><td>—</td></tr></tbody></table>

In conclusion, we study the stationary response of a resonantly pump fiber cavity through the Stokes output power as a function of the input pump power. We recall the principle of the cascaded generation of Brillouin Stokes waves in such a resonator. We have seen that the output power of a Stokes order ($N$) evolved monotonically with the pump power. Simultaneously, the output power of the pumping Stokes order ($N - 1$) is constant due to the clamping effect [178]. Thus, Stokes lines output power alternate between monotonous evolution and continuous power emission. In this section, we have shown the power dynamics of the MBL laser, which are reported in different articles [3, 34, 116]. The recall of this study helps to identify the various operating regimes of the MBL (monotonous and clamping regime). This clarification is of great importance to the following studies of RIN and frequency noise behaviors. The intensity noise dynamics of MBL will be discussed in the next section.
## 4.6 Relative intensity noise of resonant Brillouin laser

In optical communications, intensity noise affects the quality of the signal transmissions. So the intensity noise is an important parameter to characterize a laser. In resonant Brillouin laser, there are few experimental and theoretical results that are reported on RIN. A study shows a 20 dB RIN reduction at lower offset frequencies when compared to pump noise [34]. Similarly 15 dB RIN reduction was observed in higher offset frequency (2-18 GHz) [179].

This section reports one of the main contribution of this thesis regarding the intensity noise evolution of a multi-Stokes Brillouin laser. Since our objective is to determine the Brillouin laser RIN transfer function, we choose to make measurements independent of the pump RIN behavior by normalizing our Stokes RIN measurement with respect to the input pump RIN. This normalization permits to get rid of the pump-RIN features, as for example intrinsic relaxation frequency, which do not relate to the Brillouin processes themselves. This normalization simplifies the identification of RIN features. Moreover, it allows our experimental observations to be compared to numerical simulations, for which an ideal white frequency noise is considered for the pump RIN.

In the following, we will present RIN simulations and its overall evolution for the Stokes waves S1, S2, and S3, as a function of the input pump power. Afterwards, we will focus on the RIN behavior for individual Stokes lines. Then we will discuss the impact of the cavity Q-factor on the MBL RIN. Finally, the impact of gain detuning will be evaluated. All experimental RIN results will be compared to numerical simulations performed with experimentally extracted parameters.

### 4.6.1 RIN simulations

To calculate RIN numerically, we introduce noise in the pump term of Eq. (4.1) as $S = S_0 + f_r$. Here $f_r$ represents the fluctuations of the pump amplitude approximated by a Langevin white Gaussian noise source with $\langle f_r(t)f_r^*(t') \rangle = C\delta(t-t')$, where $C$ is the auto-correlation strength of $f_r$. $C$ is directly linked to the noise properties of the pump. Parameter $C$ is used as a fitting parameter. This parameter disappears after the normalization of the Stokes RIN by the input pump RIN. Numerical simulations
correspond to time streams of 7 ms with a time resolution of 3 ns. The corresponding quite large number of events justify the use of a Gaussian distribution, instead of a Poissonian one through the central limit theorem [180]. Then we take the amplitudes fluctuations $δ|A_η| = |A_η| - |A_η|_{\text{Steady}}$, and find their spectral densities $S_{\delta|A_η|}^p$ by converting them in the frequency domain and multiplying by the complex conjugated. RIN is finally determined by [4]:

$$ \text{RIN} = \frac{8S_{\delta|A_\eta|}^p}{|A_\eta|_{\text{Steady}}^2} \qquad (4.9) $$

RIN curves result from the averaging over 20 computations. Numerical and experimental RIN will be discussed in the following section.

## 4.6.2 RIN versus the cascade process

The usual measurement method of intensity noise characterization consists in the acquisition of the power spectral density of the intensity noise of the laser line. We use an electrical spectrum analyzer to obtain the PSD of the electrical signal at the detection level. The intensity noise measurement bench is explained in section (1.3). We used this technique to measure RIN of MBL.

As a first step in the study of RIN in MBL, we discuss the behavior of the RIN at 4 kHz from the carrier. An extended study at different frequencies was made in [34]. We report in figure (4.6 (right axis)) the RIN behavior of the three first individual Stokes lines as a function of the input pump power. Dotted lines correspond to measured RIN and starred lines to numerical simulations. We first focus on the S1 RIN. For a normalized input pump power of 1, corresponding to the S1 threshold, S1 RIN highlights 20 dB excess RIN with respect to that of normalized input pump. As mentioned before, the pump clamping between 1 $P_{th}$ and 4 $P_{th}$, implies a transfer of pump fluctuations towards S1 [181]. Those fluctuations are amplified through the Brillouin gain explaining the excess noise. While going away from the threshold, the S1 output power increases implying a progressive decrease of the RIN towards the input pump RIN level. It is worth to notice that, in single mode Brillouin laser, the output Stokes RIN can even be lower than the input pump RIN. The noise reduction is strongly dependent on the cavity and Brillouin gain coefficient [125], and on the laser sensitivity to environmental perturbations as in a usual laser. In the present work, we take advantage of the SBS cascade effect to reduce Stokes noise level below the input pump RIN as described in the following.
<figure><img src="image_53.png" /><figcaption>Figure 4.6: Left axis : Output Stokes power ($P_s$) normalized to the circulating power ($P_c$) versus input pump power ($P_{in}$) normalized to the Stokes 1 lasing threshold ($P_{th}$). Right axis: RIN of Stokes lines normalized to the input pump RIN. RIN amplitude is measured at 4 kHz from the carrier. Straight lines and full dots are experimental results. Stars are simulation results. Results are obtained with the "low-Q cavity" Brillouin laser, $P_{th}$ = 26.5 mW.</figcaption></figure>

Above $4P_{th}$, which corresponds to the $S2$ lasing threshold, the $S1$ experiences an abrupt RIN reduction up to 20 dB compared to the input pump RIN level. This reduction originates from the clamping effect of $S1$ due to the appearance of the stimulated $S2$ emission. In that regime, the "gain=losses" condition acts as a driving force to maintain the $S1$ output power constant (See Eq. (4.5)). Thereby, any intensity fluctuations of $S1$ are attenuated. At $8P_{th}$, coherent $S3$ emission arises. $S2$ becomes an efficient pump for $S3$, and similarly, intensity noise reduction occurs for $S2$ above the $S3$ lasing threshold power. Since $S2$ output power is clamped, $S1$ output power increases again proportionally to the input pump power. Consequences on the $S1$ RIN are instantaneous and manifest an increase of noise towards the input pump noise. Concerning the $S3$ RIN, close to the threshold, its RIN reaches 36 dB in excess to the pump laser RIN and progressively shortens towards the input pump RIN level. This high RIN level can be attributed to the noise transfer through the cascade effect. Indeed, SBS process implies energy conservation between involved waves. Then, RIN of uphill waves are transfered, and even amplified, to downhill waves. Moreover, as mentioned by Behunin [3] spontaneous anti-Stokes emission
can also degrade RIN of Stokes waves (which is not taken into account in our model).

Numerical simulations (starred lines) are superimposed on the experimental measure-
ments. RIN behavior is very well reproduced by the numerical simulations in particular:
(i) the monotonous reduction of RIN in single mode regime and (ii) the sudden reduc-
tion in the clamping regime. A discrepancy in the RIN reduction amplitude between the
model and the experiment exists in the clamping regime. The model expects up to 40 dB
reduction when the experimental results tend to -20 dB. As it will be demonstrated in
the following of this thesis, this discrepancy is due to the gain detuning of the Brillouin
laser. There is also a possible contribution coming from the anti-Stokes. Note however
this AS noise increase is screened out by the noise contribution due to detuning.

In this section, we have shown the substantial impact of the clamping effect on the
intensity noise properties of the laser. In other words, once the Stokes line labeled N + 1
reaches the lasing threshold power, the "gain=losses" condition implies the clamping of
the Stokes wave N that is playing the role of a pump. In this regime, fluctuations
of Stokes N are sharply reduced, and up to 20 dB reduction compared to the RIN of
the input pump laser is experimentally observed. Similar noise fluctuations reduction
have been demonstrated in semiconductor optical amplifier (SOA) used in the saturation
regime [182, 183, 184]. In the following section, we describe in detail the RIN behavior
of individual Stokes order for various operating point to show the rich dynamics of RIN
when MBL emission takes place.

**4.6.3 RIN of individual Stokes lines**

To get more insight on the intensity noise of individual Stokes lines, we propose to study
the complete RIN noise behavior for various input pump powers. We organized this
section into two parts. Firstly, we present RIN in the single mode regime, which is
obtained for a pump power below 4 P<sub>th</sub>. Secondly, we will explain the S1 and S2 RIN for
a pump power above 4 P<sub>th</sub>, which is called the "cascaded regime".

**4.6.3.1 Single mode regime**

The single mode regime is identified by a pumping rate between $1P_{th}$ to $4P_{th}$. In this regime, S1 wave is predominant in the cavity. In this first region, RIN decreases along with increasing pumping rate but it remains above the pump RIN level. We plot the RIN
<figure><img src="image_54.png" /><figcaption>Figure 4.7: Relative intensity noise reduction of the MBL Stokes1 when it is compared to the pump (a) Brillouin laser S1 intensity noise decreases with the injected pump power. (b) RIN of S1 normalized by the pump RIN.</figcaption></figure>

of pump and S1 between 1 kHz and 10 MHz on figure (4.7 a) for a pump power below 4 $P_{th}$. In the present study, we used a Koheras laser as the pump laser and plotted in black. Due to the relaxation frequency of pump RIN at 154 kHz, it is difficult to interpret the real dynamics of Stokes RIN at different pumping rate. It is why we normalize the Stokes RIN with respect to its pump RIN as shown in figure (4.7 b). Now the S1 transfer function is clearly seen. As a matter of fact, this approach is valid only when the laser has a RIN level in excess to the intrinsic fundamental Brillouin noise.

As we can see on figure (4.7 b), the noise of S1 is constant over a range between 1 kHz to 600 kHz; at high frequencies, it decreases with around 20 dB per decade and reaches the noise measurement limit. The RIN follows the usual noise distribution of a class B laser featuring an oscillation of relaxation. Its origin comes from the coupling between phonons and photons similarly to photons and electrons in a class B diode laser [180]. An analytical expression for the relaxation frequency in single Stokes regime ($P_{in}$ from 1 to 4 $P_{th}$) can be expressed as [4, 174]

$$f_R = \frac{1}{2\pi} \sqrt{\frac{|A_1|^2}{|A_0|^2} \times \frac{2\Gamma_B}{\tau^2} \left(\frac{2}{\tau} + \frac{\Gamma_B}{2}\right)^{-1}} \quad (4.10)$$

Figure (4.8) yields extracted frequency of relaxation from Stokes 1 in $MBL_{LQ}$ (deep blue). The results of calculations of Eq.(4.10) are plotted in lines while experimental measurements are reported as points. The relaxation frequency evolves proportionally to the square root of the pump term as for a class B diode laser [185]. It shows an excellent match between the analytical expressions and experimental results.
<figure><img src="image_55.png" /></figure>

Figure 4.8: Relaxation frequency as a function of the normalized pump power for low-Q (light blue) and high-Q (deep blue) cavities. Lines represent the results of calculation of Eq. (4.10). Error bars (±75 kHz) represent the possible misreading of relaxation frequency on RIN curve.

In conclusion, we do not observe any RIN reduction in single mode regime. In resonant Brillouin laser, cascade effect will impact the noise dynamics, which is explained in the next section.

### 4.6.3.2 Cascaded regime

The cascaded regime is obtained for a pumping rate above $4P_{th}$. In this regime, higher Stokes waves are lasing in the cavity. For sake of simplicity, this cascaded regime is classified into clamping and monotonous regimes. Clamping regime is observed for S1, between $4P_{th}$ and $8P_{th}$. Similarly, S2 will be clamped for a pump power between $8P_{th}$ and $18P_{th}$ (see figure (4.4b)). These threshold values have been largely discussed in [34]. In monotonous regime, S1 and S2 are linearly increasing with pump power. We plot in figure (4.9) the RIN of S1 (blue color) and S2 (green color) between 1 kHz and 10 MHz for these two different operating regimes: i) clamping regime (Figure 4.9 (a) and (d)) ii) and monotonous regime (Figure 4.9 (b) and (c)). We will first focus on light colored curves corresponding to the low Q-factor cavity.

In the clamping regime (Figure 4.9 (a) and (d)), a 20 dB reduction of the S1 and S2 RIN occurs at low offset frequency (< 100 kHz). As explained previously, the onset of laser emission on $S_{N+1}$ implies the clamping of $S_N$ power and then the attenuation of its RIN amplitude. In the monotonous regime (Figure 4.9 (b) and (c)) Stokes power increases monotonically with the input pump power. At low offset frequencies Stokes
<figure><img src="image_56.png" /><figcaption>Figure 4.9: Normalized RIN of Stokes 1 (a, b) and Stokes 2 (c, d) lines for various input pump power. Full and dashed lines hold for experimental and simulation results respectively. Deep and light colors refered to high ($\tau_0$ = 1.4 µs, $\tau_e$ = 3.7 µs) and low-Q factors ($\tau_0$ = 1.4 µs, $\tau_e$ = 0.5 µs) respectively.</figcaption></figure>

RIN rise to at least the pump RIN level. In this regime the noise transfer channel from "pump" to Stokes line is open. It corresponds to noise transfer from input pump towards S1 in figure (4.9 b) or from S1 towards S2 in figure (4.9 d).

For $P_{in} > 4P_{th}$ the laser shifts from single to multi-Stokes Brillouin lasing and then Eq. (4.10) is not valid anymore. The onset of S3 line for $P_{in} > 8P_{th}$ implies the appearance of multiple frequencies of relaxation originating from the complex Stokes waves coupling [3, 174]. In Figure (4.9 b), we can observe multiple relaxation frequencies from 100 kHz to 1.3 MHz due to the interaction of the photon and higher order Stokes phonons shown on the RIN of Stokes 1. Similarly, in Figure (4.9 d), the Stokes 2 has relaxation frequencies inside the interval [330 kHz, 1.4 MHz].

As shown in Figure (4.9), our experimental observations are well reproduced by numerical simulations based on our model. In particular noise reduction at low offset
<figure><img src="image_57.png" /><figcaption>Figure 4.10: Schematic presentation of frequency fluctuations converted into intensity fluctuations cavity Q-factors.</figcaption></figure>

frequencies, complex relaxation frequency shape and position and noise amplitude are
reproduced. Similar RIN behavior is observed experimentally for incoming Stokes orders
alternating sequences of low RIN level at low frequency offset when the Stokes line is
clamped and high RIN level when the noise of previous Stokes order is efficiently coupled
through SBS process. We studied the RIN dynamics of the MBL in single and cascaded
region along the pump power. Now we are going to study the influences of the cavity
parameters on RIN dynamics.

**4.6.4 Impact of photon lifetime**

In this section, we highlight the fundamental role of the cavity parameters on the RIN
properties of the MBL. For this purpose, we recall the cavity parameters, which have
been introduced in section (4.3.1). In order to obtain two different cavities, we vary
the coupling coefficients to modify the coupling lifetime τ<sub>e</sub> from τ<sub>e</sub> = 0.5 µs (low-Q) to
τ<sub>e</sub> = 3.7 µs (high-Q). The intrinsic lifetime, τ<sub>0</sub> of the resonator remains the same since
no additional losses are introduced (τ<sub>0</sub> = 1.39 µs).

As it was mentioned in the previous section, figure (4.9) reports the RIN behavior
for these two cavity configurations. The light and deep colored curves refer to the low-Q
(shorter photon lifetime) and high-Q (longer photon lifetime) resonator configurations
respectively. By comparing the RIN of both configurations, two main features may be
associated with the total photon lifetime values.

Firstly, the frequency of the relaxation resonance is directly determined by the phonon
lifetime and the total photon lifetime of the cavity as shown by Eq. (4.10). Indeed, the
shorter cavity lifetime (low-Q) systematically highlights a larger relaxation frequency for
S1 and S2 lines. This behavior has been reported in figure (4.8), showing the relaxation frequency shift versus input pump power (1 and 4 $P_{\text{th}}$) for both configurations.

Secondly, low-offset-frequency Stokes-RIN levels are systematically higher in experiments for high-Q resonator. As an example, the S1 RIN in the high-Q configuration (deep blue) presents an excess noise of 25 dB with respect to the low-Q configuration (light blue) and to the simulations (figure 4.9 b). A possible explanation can be found in the relative frequency fluctuations between the input pump laser and the cavity resonance, for which the linewidth are then of the same order. The thinner the cavity resonance is, the worse the frequency fluctuations are. They are resulting in intensity fluctuations. Figure (4.10) shows the schematic presentation of frequency fluctuations converted in to intensity fluctuations in high-Q and low-Q cavities. Resonances of the cavity are presented in red and green colors for low Q-factor and high Q-factor cavities respectively. Resonances width of the high Q-factor cavities are thin and this leads into large intensity fluctuations from small frequency fluctuations. For low Q-factor cavity, the broader resonance implies a conversion factor from frequency to intensity, which is smaller. It is why the high-Q resonator RIN is more sensitive to thermal [4] and mechanical noises [125] and favor the transfer of low frequency pump fluctuations.

Those effects are not taken into account in the present model, which explains the discrepancy in the RIN level observed at low offset frequency in figure (4.9). In the next section, we will explain the discrepancy of the experimental and numerical results in the clamping regime due to the detuning of the gain bandwidth.

### 4.6.5 Influence of detuning

In the numerical simulations presented in the previous sections, we consider a zero-detuning operation of the S1 wave, which corresponds to a match between the frequency of the density wave $\Omega$ and the frequency of the maximum of the gain $\Omega_B$. This configuration is very difficult to obtain since it implies that the SBS shift corresponds exactly to a multiple integer of the cavity FSR. Schematic presentation of gain detuning of Brillouin cavity is given in figure (4.11). In general, zero detuning condition is not fulfilled, even if a judicious choice of the pump wavelength helps to reach it, as the Brillouin shift $\Omega_B$ is dependent on the pump wavelength by $\Omega_B = 4\pi nV_A/\lambda_p$, also the SBS process is temperature dependent (see section (3.2)). To evaluate the impact of a gain detuning, we
<figure><img src="image_58.png" /><figcaption>Figure 4.11: Schematics presentation of gain detuning of Brillouin cavity.</figcaption></figure>

include in numerical simulations the gain detuning $\Delta\nu = (\Omega_B - \Omega)/2\pi$. This detuning implies that SBS Stokes waves are resonant at a frequency, which does not correspond to the frequency of the maximum Brillouin gain. Figure (4.12) focuses on the impact of the gain detuning on the RIN features of the S1 in (a) the clamped regime and (b) monotonous regime for the "low-Q cavity" configuration. Including the gain detuning allows one to fit much better the experimental results. Indeed, on figure (4.12a), we fit the S1 RIN by including a gain detuning of 500 kHz in the numerical simulations. This RIN at low frequency is much better reproduced suggesting that, even a weak detuning (500 kHz) with respect to the gain linewidth (27.5 MHz), strongly impacts the RIN-reduction performances of the S1 by 20 dB at 1 kHz. Similarly, at $9P_{th}$, by including a gain detuning of $\Delta\nu = 300$ kHz in the numerical simulations of the S1 RIN, the sharp deep profile observed around 200 kHz is attenuated and tends to that of the measurements. The discrepancy between experiment and simulations in the pump range $4P_{th}$ to $8P_{th}$ have to be attributed to the slight gain detuning that strongly limits the overall RIN-reduction performances of the MBL by up to 20 dB in the present study (see figure 4.6). These results suggest that further investigations have to be done to design optimized cavity that satisfies zero detuning conditions. In next section, we will confirm the presence of gain detuning in SBS cavity by pump-probe method.

Confirmation of detuning

A measurement setup realized to confirm the detuning of Brillouin cavity is shown on figure (4.13a). It enables to check the frequency position of a cavity resonance as a
<figure><img src="image_59.png" /><figcaption>Figure 4.12: Evolution of the Stokes 1 RIN as a function of the gain detuning for a pump power of a) 5.5 $P_{th}$ and b) 9 $P_{th}$ in the low-Q cavity configuration.</figcaption></figure>

function of a linear increase of the pump power. As we have seen in the section (2.4.2), resonances of the cavity can be measured using a probe signal. When a cavity resonance experiences gain, then this resonance will experience amplification. In case of Brillouin laser, the resonances under the gain bandwidth will experience amplification. As we have seen before, our FSR is 10 MHz, which is smaller than the gain bandwidth 27.5 MHz. If the resonance is at zero detuning condition, the central resonance at the maximum gain will be amplified; otherwise, for non-zero detuning, more than one resonance may be amplified.

The cavity is stabilized onto the pump laser-frequency for the generation of SBS. The resonances are scanned by the probe signal. In order to minimize the influence of the probe on the resonator operation, we attenuate the probe-signal power down to less than 1 mW. resolution. Firstly, we reduce the pump power and scan the resonance linewidth by the probe signal. Then we increased the pump power and measure its transfer function. These scanned signals are observed on an oscilloscope.

Figure (4.13b) shows measurements at different pump powers. At low pump power, the four identical cavity resonances are observed (black curve). When the pump power is increased, some of the resonances experiences SBS amplification. While the coupling regime is moving from under-coupling toward over-coupling regime (blue line). At higher pump power (red signal), the amplification of one resonance is clearly seen. Finally, at 14.4 mW, two modes are amplified; this shows that with this configuration, the S1 signal is not generated at the maximum gain position, because in that case, two side
<figure><img src="image_60.png" /><figcaption>Figure 4.13: Experimental setup for gain detuning measurements; Koheras laser (brown) creates Brillouin scattering (dark blue) in the cavity and the probe laser (light blue) slowly scans the Brillouin gain. b) In the inset, different optical spectra measured thanks to an oscilloscope are given for different input pump powers.</figcaption></figure>

modes would appear symmetrically around the central frequency. For the green line of the figure (4.13b), the main resonant laser frequency is thus detuned from the gain maximum. In order to reduce this detuning, the cavity length or pump wavelength has to be adjusted.

However, zero detuning is difficult to achieve for a long ring cavity. One solution is to have a servo-loop to discriminate the different modes [170]. Another solution is to implement a short cavity that can control the detuning by a pump-wavelength change. (which enables a fine control as a resolution of 1 pm will enable a control of the Brillouin shift with a resolution of 7 kHz).

In conclusion, gain detuning is an important factor. It has to be optimized for better noise performances. These studies are taken under the consideration of high pump-RIN.

4.6.6 Conclusion

In conclusion, we explicitly studied the RIN behavior of the Brillouin laser in a single mode and cascaded regime. We demonstrate that the S1 RIN is significantly reduced in the cascaded regime compare to the single-mode regime due to the clamping effect of the
pump power. Our study shows that the photon lifetime of the Brillouin cavity and a very
small change of the detuning strongly modifies the relaxation oscillation frequencies and
RIN reduction respectively. At higher pumping power, multiple relaxation oscillation
frequencies are observed due to the interaction of different set of photon and phonon
waves. These experimental observations are well supported by our numerical simulations.
Moreover, we have shown that the pump RIN noise is less significant in the resonant
Brillouin laser; it is more dependent on the cavity design and pumping power. In the
next section, the frequency noise of the Brillouin laser will be experimentally studied.

4.7 Frequency noise of resonant Brillouin laser

Low-noise laser is widely required for different applications from coherent optical com-
munications, defense, health, environment to basic sciences like optical-atomic clocks,
precision spectroscopy and quantum sensing. Frequency noise reduction of single-mode
Brillouin laser is well known and reported in literature.

In 2006, Geng et al. [126] studied the intensity and frequency noise of the resonant Brillouin fiber laser of S1. They showed a 10 dB reduction in S1 frequency noise compared to the pump frequency noise. Later in 2012, chip-based Brillouin laser was demonstrated using an ultra-high-Q micro-cavity [147]. They demonstrated that the fundamental frequency noise of S1 can reach 0.06 Hz²/Hz. A dual microcavity Brillouin laser have been realized in 2015 [186], which exhibits a white frequency noise floor at 0.3 Hz²/Hz. In this last example, the laser architecture is divided in two parts: one is a silica microdisk, which provides the Brillouin laser (S1), the second part is a large mode volume micro-rod cavity, which is used to stabilize the S1 line. This last part helps to reduce the laser flicker noise. In 2018, Brillouin laser was demonstrated on integrated Si₃N₄ waveguide platform with fundamental noise floor of 0.23 Hz²/Hz [163].

All these lasers have very low intrinsic linewidth, but integrated linewidth of these lases
are relatively large because of sensitivity to temperature fluctuations, which spreads the
linewidth. To overcome this problem, a large mode volume fiber resonator can be used
as demonstrated in [165]. The short fiber cavity suppressed the thermo-refractive fluctu-
ations. They have shown that Brillouin laser can reach 20 Hz integrated linewidth with
a fundamental noise floor of 2 Hz²/Hz. Theoretical analysis of S1 Brillouin laser noise
dynamics under 4 P<sub>th</sub> (single-mode regime) was studied in [4]. All the above mentioned
experimental observations are below $4 P_{th}$ (single-mode regime) pump power, and are in agreement with theoretical results [4]. About the multi-Stokes Brillouin laser, there are very few experimental results on frequency noise dynamics.

In this section, we will explain the main features of the frequency noise dynamics of single-mode and cascaded regime of the MBL. In a population inversion laser, coherency is provided by the effect of resonance inside the cavity. The gain medium does not preserve the coherency of the laser pump. In a Brillouin laser, however, the initial coherence of the pump laser is transmitted or even improved by the nonlinear process. The initial phase of the pump laser is redistributed on the Stokes and anti-Stokes waves. There is a strong relationship between the phase of the pump laser and the phase of the Brillouin laser obtained by nonlinear interactions between the pump wave, the Stokes wave and the acoustic wave propagating in the fiber [34]. As we have seen in non-resonant Brillouin laser, frequency noise is strongly dependent on cavity losses, acoustic damping rate and initial pump phase noise. But in the case of resonant Brillouin laser, frequency noise depends not only on the acoustic damping rate and cavity losses, but also on the detuning of the gain bandwidth, and phase noise carried from spontaneous anti-Stokes. Detuning of the gain bandwidth will enhance noise coupling between amplitude and phase fluctuations. Another noise transferring channel is the cascading effect; the resonant cavity can make multiple Stokes orders, which allows the re-injection of the spontaneous anti-Stokes ($N+2$) on Stokes (N). This increases the phase noise of Stokes wave. It will be discussed in the coming sections. As we will show in the following, we have studied the dynamics of MBL above $4 P_{th}$ in the cascaded region.

As we have seen in chapter 3, non-resonant Brillouin laser frequency noise is strongly dependent on cavity losses and acoustic damping rate and initial phase noise of the pump. In the case of resonant Brillouin laser, frequency noise is also dependent on the detuning between the gain maximum and the resonant frequency, which will enhance noise coupling between amplitude and phase fluctuations as shown in section (4.6). The detuning is thus an important factor to control in order to optimize the noise performances of the Brillouin laser. Zero detuning is difficult to obtain in a long cavity; so we build a short cavity for the study of frequency noise, which allows us to control the detuning by changing the pump wavelength. Our experimental setup will be explained in section (4.7.1). We will then discuss the frequency noise dynamics of individual Stokes lines and their dependence on the pump power. In particular we will demonstrate experimentally, as far as we know
for the first time, recent theoretical predictions [3] that underline the role of anti-Stokes
waves, which contribute to increase the phase noise level of laser Stokes lines. Moreover,
we will study the influence of the Q-factor and detuning of the gain bandwidth on the
frequency noise of the Stokes waves.

4.7.1 Short Brillouin cavity

We have seen the impact of the detuning on the RIN dynamics of MBL. It is thus
important to reduce its contribution for better noise performances. It is why we develop
a short cavity for frequency noise study. In the following, the laser cavity is detailed as
well as the procedure to find the minimum detuning condition. Brillouin laser is generated
by a tunable external cavity laser (Tunics OM), of which the signal is injected into a high-
Q optical fiber resonator. The resonator consists of an optical coupler with the two arms
spliced together. The effective length of the resonator is 2 meters and corresponds to
an FSR of 100 MHz. This ensure a single mode operation since the gain bandwidth is
$27.5 \pm 2$ MHz.

<figure><img src="image_61.png" /><figcaption>Figure 4.14: Experimental setup of the BFL cavity with PDH locking.</figcaption></figure>

Figure (4.14) shows the experimental setup of Brillouin laser. Pump laser is stabilized on a cavity resonance by a PDH locking, which reduces the laser flicker noise as we discussed before in section (1.5.2). Pump laser frequency is phase modulated by radio frequency signal at a frequency of 20 MHz, which is much higher than the cavity response. The phase modulator (PM) will generate two side-band frequencies at 20 MHz. Cavity
transmitted signals are composed of resonant and off-resonant signals. The detected sig-
nals from the cavity produce a phase difference between the incident and cavity-stored
signals through the beating between the carrier frequency and its side-bands. The signal
is then demodulated with the same frequency, which is applied onto the phase modula-
tor. The correction signal is sent to the laser current through a PID. The laser carrier-
frequency follows the deviation of the cavity-resonance frequencies. Then it improves the
light coupling into the cavity and generates a single Stokes wave.

<figure><img src="image_62.png" /><figcaption>Figure 4.15: BFL detuning measurement: The measured Brillouin frequency shift (squares) and S1 threshold (dot) are plotted as a function of pump wavelength.</figcaption></figure>

As mentioned before, the Brillouin laser efficiency is improved when the Brillouin shift
is proportional to the cavity FSR. However it does not imply that the Stokes frequency
is at the gain-maximum frequency due to temperature dependence of the Brillouin shift.
In order to obtain this desired condition, the Brillouin shift may be adjusted by changing
the pump wavelength or the cavity resonance thanks to a PZT. Reaching this condition
of zero detuning, will also reduce the laser threshold. In the following, the detuning will
be controlled by a change in the pump wavelength.
When the laser reaches a minimum threshold, we can conclude that the gain detuning
is minimized ($\nu_B = n \times \Delta\nu_{FSR}$). The coupling ratio of the cavity is set at the critical
coupling. Then the pump wavelength is varied from 1538 nm to 1548 nm. Simultaneously,
the S1 threshold and the Brillouin frequency shift are measured. Figure (4.15) shows the
experimental results of the threshold (black circles) and Brillouin frequency shift (red
square dots) as a function of the pump wavelength. This Brillouin frequency shift is
linear with a slope coefficient of $\frac{d\Omega_B}{d\lambda_P} = -2\pi \times 7\,\text{MHz/nm}$. The minimum threshold is
14.78 mW for the critical coupling. When changing the pump wavelength, the evolution
of the S1 threshold is parabolic, which confirms the Brillouin gain bandwidth of 27.5
MHz [147] (1 nm corresponds to 7 MHz; the whole scan over 10 nm to 70 MHz).

From this analysis, we operated the pump laser at 1544 nm for minimizing the de-
tuning in the following frequency noise studies. We have seen before in section (3.6) that
Brillouin laser noise is dependent on the pump noise in non-resonant cavity. In resonant
cavity it is also only dependent on the gain detuning, pump power and spontaneous anti-
Stokes as it will be explained in the following section. For the sake of simplicity, firstly we
will explain, in section (4.7.2), the influences of anti-Stokes spontaneous scattering on S1
frequency noise dynamics. Then we will detail influences of gain detuning on frequency
noise spectrum of S1 and S2 in section (4.7.3). Finally, we will study the impact of photon
lifetime in noise reduction and we will compare our results with other Brillouin lasers in
section (4.7.4).

4.7.2 Frequency noise versus the cascaded process

Objective of this section is to detail the frequency noise dynamics of Brillouin laser in
presences of higher order Stokes inside the cavity. We will explain the role of anti-Stokes
generation in cascaded process and the associated enhancement of frequency noise.

4.7.2.1 Role of anti-Stokes wave

Recent theoretical results [3] have shown that the reduction of frequency noise will be
affected by the coexistence of anti-Stokes waves in resonant Brillouin laser. The schematic
picture of anti-Stokes generation in MBL is shown in figure (4.16). The x-axis is for the
frequency and the dotted line are the cavity resonances, which are separated by the
FSR ($\Delta\nu_{FSR}$). When the pump frequency (in red) is locked onto a cavity resonance, it
<figure><img src="image_63.png" /><figcaption>Figure 4.16: Schematic of anti-Stokes generations in cascaded Brillouin laser; dotted line : resonance of the cavity; shaded spectrum are Brillouin gain profiles for Stokes and anti-Stokes: S1 (blue), S2 (green), S3 (yellow); sharp peaks are Stokes stimulated laser lines and pump (green for S2, blue for S1, red for the pump).</figcaption></figure>

will interact with thermally agitated phonons and it will create spontaneous Brillouin scattering. This spontaneous scattering will generate the Stokes (S1) and anti-Stokes (AS1) in the cavity (blue spectrum) [77]. When the pump power increases above the S1 threshold, the Brillouin process enters into the stimulated regime; note that to create one anti-Stokes photon, both a pump photon and a phonon have to be virtually absorbed [78]. This last condition lowers the efficiency of the anti-Stokes generation. Recall that for the generation of a Stokes photon, a pump photon is virtually absorbed and a phonon is created. Similarly, when the S1 power reaches $P_{th}$, spontaneous Stokes (S2) and anti-Stokes (AS2) are generated with a frequency respectively downward and upward shifted with respect to that of S1 (figure (4.16)). The upward shifted anti-Stokes (AS2) frequency is at the pump resonance frequency (small green spectrum). Similarly, when S2 reaches the threshold, which means at $4 P_{th}$, spontaneous Stokes (S3) and anti-Stokes (AS3) are created (small yellow spectrum). This anti-Stokes wave will enhance the S1 frequency and intensity noise in the cascaded process (figure (4.17) extracted from reference [3]). In order to confirm the influences of anti-Stokes, we will study the frequency noise of S1 as a function of the input pump power in the following section. Recall that the intensity noise increase is screened out by the detuning effect. We clearly show that there is an effect on frequency noise.

### 4.7.2.2 Experimental results

The cavity is pumped by a tunable laser at 1544 nm, and the pump frequency is stabilized onto one cavity resonance using a PDH locking. The coupling ratio is fixed at 2% (light coupled inside the cavity) and gives a threshold of 15 mW when detuning between Brillouin gain maximum and S1 resonance is minimum (figure (4.15)). The intrinsic
<figure><img src="image_64.png" /></figure>

Figure 4.17: Theoretical results from [3] showing the degradation of linewidth of different Stokes components due to AS. This degradation can be equivalently observed in frequency noise (chapter 1, section 1.4.2).

photon lifetime is then $\tau_0 = 0.29$ µs, which is equivalent to a Q-factor of 129 millions. The MBL S1 output is filtered using a 6 GHz bandwidth optical filter. As we have seen in section (1.4), the frequency noise of S1 is characterized using a fiber-based Mach-Zehnder Interferometer, which acts as an optical frequency discriminator. The experimental setup was explained in section (1.4.3). Here, we used a 200-meter fiber as a delay line for measuring the S1 frequency noise, which corresponds to a free spectral range of 1 MHz. The output of the MZI is detected and measured using photodetector and signal sources analyzer (FSW-13 GHz) respectively.

<figure><img src="image_65.png" /></figure>

Figure 4.18: Cascaded Brillouin laser frequency noise dynamics (a) S1 frequency noise at different pump powers in linear scale (b) S1 white noise at a fixed frequency of 200 kHz as a function of normalized pump power.

Figure (4.18a) shows the measured frequency noise of the pump and S1. The frequency noise of S1 is measured for different input pump powers. The interest is for the laser white noise. As we have seen before the Tunics OM laser has a white frequency noise of
$2 \times 10^3 \text{ Hz}^2/\text{Hz}$, which corresponds to an intrinsic linewidth of 6.28 kHz. Pump laser free-running (FR) is shown in black color and the frequency-locked (FL) pump laser is plotted in gray color, both lasers experience the same white noise level. S1 frequency noise is shown at different pumping rates from 2.3 $P_{th}$ to 11.3 $P_{th}$. Experimental results reveal again the expected significant reduction in S1 frequency noise when compared to that of the pump noise; it also clearly demonstrates that reduction is varying with the input pump power. The white frequency noise of S1 reaches $4 \text{ Hz}^2/\text{Hz}$ at 3.8 $P_{th}$ pumping power; this gives a 26 dB reduction in frequency noise when compared to the pump noise level.

Figure (4.18b) shows the frequency noise level of S1 at a fixed frequency of 200 kHz as a function of the normalized input pump power. The S1 frequency noise decreases exponentially with the input pump power when the cavity is below the S2 threshold. A decrease of one decade corresponds to a variation of the normalized pump of $1.5 = 22.2 \text{ mW}$. At a normalized pump power of 2.32 corresponding to 34.8 mW ($2.32 \times 15$), the noise level is $27.8 \text{ Hz}^2/\text{Hz}$; at 3.8 (57 mW) the noise level is $4.09 \text{ Hz}^2/\text{Hz}$. Once, it reaches the S2 threshold, frequency noise increases sharply from 4 $P_{th}$ to 8 $P_{th}$. As explained before, for pump-power between 4 $P_{th}$ and 8 $P_{th}$, S1 experiences a clamping effect in its output power, which lowers its noise. However an extra noise source is injected through the spontaneous anti-Stokes scattering AS3 due to the S2 Stokes wave. That will increase the S1 noise and will decrease the reduction factor "pump linewidth over S1 linewidth". Note that as shown in figure (4.16), the AS2 upward frequency (small green spectrum) is at the pump frequency while AS3 frequency (small yellow spectrum) is at the S1 downward frequency. The AS3 photons are feeding the S1 frequency and will absorb phonons, that are necessary for this AS3 generation. It leads to a frequency noise increase in the 4 $P_{th}$ to 8 $P_{th}$ region. Similarly, when S1 reaches 8 $P_{th}$, the frequency noise is decreasing inversely along with the pump power because S2 reaches the clamping regime that reduces the anti-Stokes presence in the cavity, which implies a noise reduction of S1 above 8 $P_{th}$. This result shows that the cascaded process may degrade the S1 linewidth up to one order of magnitude in presence of higher order Stokes in the cavity. Also, to the best of our knowledge, it is the first experimental validation of this theoretical study [3]. In the following section, we will explain the frequency noise spectrum of the Stokes waves.
4.7.3 Frequency noise of individual Stokes lines

In this section, we will study the individual Stokes signals frequency noise. The objectives
of this section are to discuss the main advantages and limitations of noise reduction in
MBL, at the view that we are now able to master the detuning between S1 and the
gain maximum. The main problem in such analysis is the fact that the noise floor is a
limitation that is quickly reached. To overcome this limitation, in the following studies
we will use a low-Q cavity.

4.7.3.1 Analysis of low-Q cavity

A low-Q cavity will decrease the linewidth reduction and will permit to get a S2 frequency noise above the noise floor. It is why we investigate frequency noise of MBL by changing the coupling ratio. This allows us to increase the output power of S2. The cavity is set into under-coupling with an extrinsic photon lifetime of τ<sub>e</sub> = 0.15 µs, and this implies that 13 % of the light is coupled into the cavity, leading to a decrease of Q-factor of 59.8 millions. The S1 MBL threshold is increased to 31.6 mW. To measure the frequency noise of the MBL laser, we use the same frequency noise characterization bench based on an MZI. However, to enlarge the measurement bandwidth up to 20 MHz, we reduce the branch asymmetry by using a fiber spool of 10 meters. This configuration allows the observation of detuning effects in Stokes lines frequency noise.

4.7.3.2 Analysis of S1 and S2 lines

Figure (4.19a), shows the single-mode and clamped regions of S1 frequency noises in the low-Q cavity configuration. As we discussed before in section (1.5.2), by doing PDH locking on a cavity resonance, pump low-frequency-noise can be reduced. In free-running operation, the laser pump exhibits a 186 ± 18 kHz integrated linewidth (with 10 ms observation time). By locking the pump laser frequency onto a cavity resonance, the flicker noise is reduced and corresponding the integrated linewidth was improved down to 6.28 ± 0.6 kHz for 10 ms observation time. Moreover, the SBS laser will again reduces flicker frequency noise then the S1 integrated linewidth is improved to a value lower than 2 kHz. However, the measured flicker frequency noise is limited by the noise floor of the measurement bench below 8 kHz Fourier frequency (see figure (4.19) gray dotted lines).
<figure><img src="image_66.png" /><figcaption>Figure 4.19: (a) S1 frequency noise (b) S2 frequency noise for different regimes.</figcaption></figure>

We are now interested in analyzing the white noise of the Brillouin laser. Below 4 $P_{th}$ (single-mode regime), the frequency noise of S1 (shown in red color in figure (4.19a)) has a white frequency noise of $20 \text{ Hz}^2/\text{Hz}$. The peak at 2.6 MHz can be explained by the theoretical results of Loh et al. [4] and will be discussed in the next section. Above 4 $P_{th}$ (cascaded regime), S1 white frequency noise is increased due to an excess noise of spontaneous anti-Stokes S3. Similarly, S2 frequency noise is shown in figure (4.19b) at different pumping rates. Close to threshold, S2 frequency noise is limited to $2 \times 10^2 \text{ Hz}^2/\text{Hz}$ at 100 kHz. Along with increasing pump power, the S2 noise decreases by 3 dB down to $10 \text{ Hz}^2/\text{Hz}$ at 7.7 $P_{th}$ with a resonance frequency located close to 4.5 MHz. Close to the S3 threshold, the frequency noise of S2 reaches $4 \text{ Hz}^2/\text{Hz}$. The resonance peak in the experimental results will be explained in the next section.

### 4.7.3.3 Transfer function

The transfer function of pump frequency noise to Stokes frequency noise is shown in figure (4.20a) for a pump power of 1 mW with no detuning. The major trends of this transfer function were underlined in [4, 123]. At low frequency, it is constant up to a resonance frequency of 5 MHz, which corresponds to the relaxation frequency of the cavity. At higher frequencies, noise transfer decays with a -40 dB/decade slope. Similarly, figure (4.20b) shows the transfer of frequency noise from the pump to the Stokes when the pump frequency and SBS gain are detuned from the resonance [4]. At low frequencies, the pump noise to Stokes transfer function is constant like the zero detuning configuration. At higher frequencies the system has a sharp resonance near 5 MHz when the detuning is
nonzero. This resonance peak may be due to an amplitude-phase coupling between the pump amplitude noise and the the pump resonant frequency noise, of which the phase of the Stokes signal is related [4]. When the resonance is detuned from the gain maximum, then this noise transfer will be enhanced.

<figure><img src="image_67.png" /><figcaption>Figure 4.20: The pump frequeny noise to Stokes frequency noise (a) zero detuning (b) with pump and SBS gain detuning [4].</figcaption></figure>

The resonance frequency shifts in the figure (4.19) may be then explained by the pump power and detuning dependencies of the relaxation frequency. We have shown that the model that we introduce is able to perfectly match the experimental and theoretical RIN by adjusting the detuning and by inference the relaxation peak and its dependencies to detuning and pump power (report to figure (4.12), Eq. (4.10) for $P_{th} < P_{in} < 4 P_{th}$). Thus the observation of the peak shift in the frequency noise when the detuning is nonzero along with the pump power may be interpreted at the view of our results on the RIN evolution with pump power and detuning. First of all, when the power is increased, there is a temperature increase inside the cavity that will imply a modification of cavity length and then of Brillouin-frequency shift and of detuning. Secondly if we made the assumption that there is an amplitude-phase coupling between intensity and phase noise, then the dependence of the relaxation frequency peak in the frequency-noise spectra may be understood by our results on RIN dependence. Thirdly, the peak in the the RIN spectra is a very complex function of power and detuning that could be simplified in simple cases (for instance Eq. (4.10) for $P_{th} < P_{in} < 4 P_{th}$). It is perfectly simulated by our model that we have introduced and it can quantitatively explain the evolution of the resonant peak.
4.7.3.4 Frequency noise comparison of S1, S2

Figure (4.21) compares the frequency noise of S1 and S2 at the same pumping rate. We have seen, as expected that the noise of S1 is higher compared to that of S2 when the pump power is close to the S3 threshold. S1 frequency noise experiences an excess noise from the spontaneous anti-Stokes of S3 at 8.3 P<sub>th</sub>. Also, a large resonance peak is obtained at 5 MHz due to a strong coupling between amplitude and phase noise, which confirms that cavity is no longer at zero detuning [4, 174]. Since the linewidth and frequency noise of a laser are closely related (section (1.4)), we observed that the Lorentzian linewidth of S1 reaches 125.6 Hz, S1 is acting as a pump for S2, which reduces the linewidth down to 12.56 Hz. In figure (4.21), the dotted lines gives the noise floors for the system, which was measured using a balance MZI with 10 m fiber in both arm of the MZI ( the experimental setup is detailed in the section (1.4.4)). Above 20 kHz, S1 and S2 frequency noise are above the noise floor of the measurement bench. Below the 20 kHz, the measurement is suffering noise from the fiber in MZI. Unfortunately, the measurements are then limited below 20 kHz from the carrier. However it can be clearly seen that the frequency noise at the resonant peak is the same for S1 and S2, which is clearly a limitation in the reduction process, and that in the range [20 kHz, 300 kHz], the reduction ratio S1-pump frequency noise is around 16.9 dB while 10 dB is reached in the reduction ratio S2-S1 frequency noise. The ideal case will report a same reduction in the cascaded process, which is a nice way to reduce the frequency noise by operating on higher-order Stokes waves. As it was underlined in the Fresnel's thesis [34], the Brillouin shift is dependent on the Stokes order as it is dependent on the wavelength. It is why if a zero detuning is realized between

<figure><img src="image_68.png" /><figcaption>Figure 4.21: Comparison of frequency noise of S1 and S2 at 8.3 P<sub>th</sub></figcaption></figure>
the pump and S1 in order to get the best performances in terms of noise as shown in figure (4.21), the detuning cannot be zero for the higher order Stokes (See appendix D). This will degrade the performances of frequency-noise reduction for higher orders. Note that the Brillouin shift is also dependent on the acoustical velocity in the medium. At constant pressure and temperature, it is constant.

### 4.7.3.5 Conclusion

We have experimentally studied the MBL frequency noise dynamics in single-mode and cascaded regimes. The Stokes frequency noise is filtered by the SBS process. We have implemented a locking, which reduces the flicker-noise of the pump laser. By using the combination of PDH and SBS process, we have achieved a low noise laser in terms of short and long term stability. Also, we have experimentally shown that spontaneous anti-Stokes of (N + 2) enhances the noise of N th order Stokes, which confirms the theoretical prediction of [3]. Gain detuning induces additional noise to the Stokes signal [4, 174]. Also, we have underlined that the detuning effect is inherent to the cascaded laser. In the next sections, we are studying the impact of photon lifetime on the frequency noise of the S1.

### 4.7.4 Impact of photon lifetime

In this section, we will highlight the role of cavity parameters on the frequency noise of the Brillouin laser. For this study, we will compare the S1 frequency noise of two cavities with different coupling coefficients at 3.8 $P_{th}$. Cavity Q-factor is tuned by the coupling coefficient, as specified before; one is over-coupled with a Q-factor of 129 million, and the other under coupling regime with a Q-factor of 59.8 millions. Figure (4.22) reports the frequency noise of the S1 and the pump for the two-cavity configurations. The green and blue colored curves refer to high-Q and low-Q resonator configurations, respectively. The S1 frequency noise is sharply reduced in both configurations due to the combined influences of the photon decay rate and acoustical damping [123]. By increasing the photon lifetime in the cavity, the frequency noise reduction is improved by 6.9 dB from 20 dB to 26.9 dB, which is in agreement with theoretical values estimated using Eq. (3.5). In low-Q cavity, the estimated reduction factor is 20.5 dB, for a high-Q cavity it is 26.4 dB. This equation is only valid in single-mode SBS laser, especially at $4P_{th}$ above this
pump power, other noise enhancement channels are opened. In this example, S1 reaches a Lorentzian linewidth of 62.8 Hz for low-Q and 12.5 Hz for high-Q cavity configurations.

<figure><img src="image_69.png" /><figcaption>Figure 4.22: Impact of the photon lifetime in S1 frequency noise at a given pumping rate</figcaption></figure>

In this thesis, we are studying the dynamics of the Brillouin laser noise, more than the ultimate noise limit. Even though it is interesting to compare our S1 frequency noise to reported Brillouin laser frequency noises in single mode regime. Table (4.3) sums up some of Brillouin laser linewidths from different articles.

<table><caption>Table 4.3: Comparison of S1 linewidth in different cavities</caption><thead><tr><th>Cavity</th><th>Q-factor</th><th>Pump linewidth (Hz)</th><th>S1 linewidth (Hz)</th><th>Reduction (dB)</th><th>Frequency noise @ 100 Hz (Hz<sup>2</sup>/Hz)</th><th>Output power @ 4P<sub>th</sub> (mW)</th></tr></thead><tbody><tr><td>Ultrahigh-Q wedge disk (silica on silicon)[147]</td><td>1.4 × 10<sup>8</sup></td><td>942</td><td>0.18</td><td>37</td><td>10<sup>5</sup></td><td>0.035</td></tr><tr><td>Si<sub>3</sub>N<sub>4</sub> chip [163]</td><td>2.8 × 10<sup>7</sup></td><td>163</td><td>0.72</td><td>23.5</td><td>10<sup>7</sup></td><td>20</td></tr><tr><td>2 m Silica fiber (our)</td><td>1.3 × 10<sup>8</sup></td><td>6280</td><td>12.5</td><td>26.9</td><td>5 × 10<sup>3</sup></td><td>25</td></tr></tbody></table>

We are comparing three different Brillouin lasers: the first one is made of a silicon micro-disk, which gives a fundamental linewidth of 0.18 Hz with 37 dB reduction compared to that of the pump laser. The drawback of this laser is a very small S1 output power (micro watts) and high flicker noises. The second one is a Si₃N₄ integrated chip Brillouin laser with a fundamental linewidth of 0.72 Hz. Output power of the this laser is of the order of 10's of mW. The main disadvantage of this laser is a large flicker noise due to therm refractive noise. For our laser, S1 linewidth is much larger compared to
presented value in table (4.3). Indeed, as stated in this thesis, the linewidth is strongly
dependent on the pump laser linewidth. It turns out that our pump laser has a larger
linewidth (6.28 kHz) when compared to those used in table (4.3). However the reduction
factor is better than the integrated chip Brillouin laser. Even-though we were not able to
estimate the integrated linewidth as the noise floor is reached below 20 kHz. Estimated
integrate linewidth is 2 kHz at 10 ms. A similar architecture fiber Brillouin laser shows
that S1 linewidth can reaches 20 Hz with a simple PDH lock, also it can be further re-
duced to 2 Hz by suppressing thermal fluctuations [165]. These results shows that fiber
based Brillouin laser have great potential to realize stable single frequency lasers.

4.8 Conclusion

We have explained cascaded Brillouin fiber laser, in terms of noise and power dynamics
by comparing experimental and analytical results. Theoretical results are obtained with
the help of the modified coupled-mode equations. We define two regimes of operation
depending on the pump power: single-mode regime and clamping regime, showing that
low offset frequency RIN of considered Stokes lines can be strongly attenuated when op-
erating in the clamping regime when pump RIN is larger. When working in a single-mode
regime, our study of the impact of photon lifetime on the relaxation frequency shows an
analogy between Brillouin fiber laser and Class B laser behavior. Experimental observa-
tions are well supported by numerical simulations that reproduced the main features of
multi-Stokes RIN. Moreover, we have analyzed the significant influences of the gain de-
tuning, which degrades the multi-Stokes RIN. Similarly, we studied the frequency noise
dynamics in the cascaded region, which give nontrivial power dynamics; this behavior
originated from the new noise channel open by the cascaded process. The experimen-
tal results have shown that this noise channel increases the linewidth of S1 at a higher
pumping rate. However, the cascaded Brillouin laser exhibits excellent noise reduction
in frequency. We reach a linewidth of the 12.5 Hz with a 129 millions Q-factor cavity.
This results can be improved by using a low noise pump sources and high-Q cavity. Also,
we demonstrated that the resonant Brillouin configuration suffers from noise coupling
between amplitude and phase when it is detuned from the maximum of the gain band-
width. From these experiments, we observed that SBS laser has excellent performances in
frequency noise; its RIN characteristics appear worse compared to semiconductor lasers.
Our study gives inputs to design optimized Brillouin fiber laser in terms of RIN and
frequency noise reduction.
<figure><img src="image_70.png" /></figure>
# General Conclusion

My Ph.D work is devoted to the understanding of noise dynamics in Brillouin fiber-ring lasers.

The first chapter of this manuscript presents generalities about laser noise. The relative intensity noise of lasers is theoretically explained and experimentally measured using a commercial DFB laser (KOHERAS). As well, the frequency noise of lasers is explained and related to the laser linewidth. An illustration is given thanks to a tunable laser (Tunics OM) and the linewidth is measured using the so-called *β-line* technique, which is compared to a self-heterodyne technique. One important characterization of such noise qualifications is the evaluation of the noise floor of the bench and its analysis. This analysis was carried out using different delay lengths in the interferometer, which is used as an intensity converter for frequency fluctuations. This was discussed in details. In addition, in order to obtain long-term stability (on the order of a few hours) and to facilitate our measurements, we have demonstrated the stabilization of a laser on a fiber optic resonator using the Pound-Drever-Hall (PDH) technique. This technique allowed us to reduce the line width of the laser from 203 kHz down to 2 kHz.

In the second chapter, we have presented a general introduction of spontaneous and stimulated Brillouin scatterings. We have discussed and measured the key parameters of Brillouin scatterings, which are gain bandwidth, frequency shift, and Brillouin gain coefficient (maximum gain). A new Brillouin gain characterization method based on cavity ring-down method (CRDM) was introduced. The cavity characterization using CRDM, which helps to identify the coupling regime and intracavity losses of the resonator was also explained. Finally, we showed that the estimated Brillouin gain coefficient using CRDM is in good agreement with the results obtained from the self-heterodyne method. The CRDM method offers the possibility to obtain the material gain and cavity parameters, in a single experiment, using optical pump powers of the order of tens of
milliWatts, for cavities of short length (meter, millimeter, micrometer).

In the third chapter, we reviewed Brillouin laser development over the last few decades. This ranges from fiber ring laser to integrated chip-based Brillouin lasers. After explaining the difference between a resonant-pumped and non-resonant-pumped Brillouin laser, the single-mode or multimode character of the Brillouin laser operation is explained. Mod- hops of the non-resonant Brillouin laser are demonstrated using a 20-meter Brillouin cavity. We showed the suppression of mode hopping using a PDH locking technique. It allowed us to obtain a non-resonant Brillouin laser with long-term stability (~ hours). Moreover we are able to control key parameters such as cavity factors (Q-factor, coupling losses, internal losses), the detuning between the laser Stokes resonance and the gain maximum (or equivalently the Brillouin shift), pump noise. All these improvements allowed us to have a fine control of the laser, which has a long term stability, and to make a qualitative and quantitative comparison between theory and experiments. The analysis in this third chapter is made using standard model of noise transfer for non resonant Brillouin lasers. We demonstrated that our system can reach intrinsic linewidth of 1.3 Hz with a 37 ± 1 dB reduction when compared to the pump laser linewidth. Furthermore, we also showed that the intrinsic linewidth of the Brillouin laser can reach the level of mHz by cascading of two non-resonant Brillouin lasers.

In the last chapter, the noise dynamics of the resonant Brillouin laser is investigated experimentally using a fiber ring cavity. A model generalized to multi-Stokes Brillouin lasers was developed at Foton Institute; We have shown that our experimental measurements are in good agreement with the numerical simulations for the Brillouin laser. The RIN of Brillouin lasers with multi-Stokes orders was then experimentally and numerically studied in terms of pumping power, photon lifetime and gain detuning. We have confirmed that in the cascade regime of MBL, and for certain frequency ranges, the RIN can be reduced compared to single-mode regimes. This reduction is due to the saturation of the intracavity pump power due to the lasing process.

The photon cavity lifetime changes mainly the MBL relaxation frequency.

The detuning between the laser Stokes resonance and the Brillouin gain maximum has a great influence on the response of the RIN, even for small values (~ some 100 kHz). We detail theoretically and experimentally the increase of RIN by the influence of gain detuning.

Finally we tried to explore the limitations in frequency noise reduction using two
Brillouin lasers. We have shown that it is possible to cascade frequency-noise reduction by cascading two Brillouin lasers. However our bench noise floor is rapidly reached and prevented us to explore experimentally any limitation in noise reduction even if the thermal noise will be in any case a fundamental limitation. Nevertheless intrinsic linewidth of the order of 100 mHz is easy to get in a single-Stokes configuration or in a multi-Stokes configuration.

Other techniques are currently being explored to overcome the noise floor limitations of our frequency noise measurement benches in order to allow us to reach the fundamental limits of noise reduction for the intrinsic linewidth. It goes without saying that the low frequency noise (flicker noise, random walk) associated to thermal and mechanical fluctuations necessitates a slaving on a metrological reference to achieve small integrated linewidths. We succeeded to reach the kHz integrated linewidth using, as a reference, the resonance frequency of an fibered interferometer.

However, we have pointed out that there are fundamental limitations for Brillouins lasers with several Stokes orders. We have experimentally demonstrated, in agreement with theoretical predictions showing the penalty caused by anti-Stokes waves, that a degradation of frequency noise is observed. This can be lowered by operating far from the threshold of the analyzed Stokes order. Moreover the Brillouin shift is dependent on the Stokes order as its frequency shift is wavelength dependent. That will generate an inherent detuning when the Stokes order is increased. As precisely shown in this thesis, this detuning is degrading the RIN and the frequency noise by amplitude-phase coupling within the nonlinear process. It turns out that it is a severe limitation for Brillouin lasers with several Stokes orders, as a small detuning (less than 1 MHz) will affect both RIN and frequency noise.

The bad news is that controlling such a detuning on the MHz range seems to be out of reach of a low cost system, as it was expected in the project FUI SOLBO with partners IDIL, Lumibird, THALES, Photonics Bretagne, ISCR. The good news is that this work allows the engineering and sizing of these Brillouin laser systems and whether it can meet specifications in terms of RIN and frequency noise. Despite these limitations, interesting performances (Hz, 100 mHz intrinsic linewidth, kHz integrated linewidth) could be reached by designing properly the laser system.

As shown by many very recent publications on this topic, many groups around the world have understood the interest of Brillouin lasers to get very coherent wave. This
work is a contribution to the understanding of Brillouin fiber lasers with multi-Stokes orders.
# Appendix A

## Fresnel representation of laser noise

In figure (A.1a) shows the Fresnel representation of the complex field $E(t)$ of the laser signal. The initial magnitude of field is $E_0(t)$. Then the addition of a noise event, having magnitude of $\delta E(t)$ due to spontaneous emission process. This will results in perturbation of both amplitude and phase of the original oscillating field.

The parallel component $\delta E_\parallel(t)$ models the variation of the amplitude of the field $\delta E(t)$. In a laser, such a variation will be systematically corrected by the phenomenon of gain saturation relative to the amplifying medium. This phenomenon decreases any increase in the amplitude of the field, so the amplitude fluctuations caused by the spontaneous emission are reduce. The gain saturation phenomenon therefore acts as a restoring force that tends to stabilize the amplitude of the wave.

<figure><img src="image_71.png" /><figcaption>Figure A.1: (a) Schematic representation of spontaneous emission of laser field (b) Contribution of stimulated emission generated by optical amplification and filtered by an optical cavity.</figcaption></figure>
The variation of the phase is modeled by the component $\delta E_{\perp}(t)$, perpendicular to the field $E_0(t)$. Unlike intensity variation, no phenomenon equivalent to that of gain saturation exists to damp a phase variation in a laser. It is therefore understandable that the phase variations will accumulate thus making the phase of the field evolve according to a diffusion process. The phase of a laser follows the process of a random walk. These phase fluctuations prevent the power spectrum of a laser from having an infinitely linewidth (i.e., zero) but rather from a fundamental linewidth.

Figure (A.1b) gives an image of the Fresnel vector of the laser field, in a rotating at the average optical pulsation of the laser field filtered by an optical cavity. The uncertainty on the field is represented by a ring arc. As has been explained previously, the radial component gives the intensity noise generated and attenuated by the saturation of the laser gain. The tangential component gives the phase noise. This arc is moving erratically due to temperature or acoustical noise on slow dynamics. It could take minutes to cover a portion of the circle by a slow erratic drift of the arc with a stochastic signal containing many frequencies. The angle arc is actually limited because it is related to the fineness of the hot cavity, typically $10^5$ for a DFB semiconductor laser, given for a 1 MHz linewidth and a free spectral range of 100 GHz. In fact, a flicker noise would lead to a random rotation of this Fresnel vector (with its ring arc) in a low frequency range.
# Appendix B

## $g_{th}$ - factor

The analytical equation for threshold and its detailed derivations can be found in [86]

$$P_{th} = \frac{g_{th} A_{eff}}{k_{polar} g_{B} L_{eff}} \qquad (B.1)$$

$$g_{th} = \Psi \left( 1 + \frac{\frac{3}{2} \ln(\Psi)}{\Psi - \frac{3}{2}} \right) \qquad (B.2)$$

$$\Psi = -\ln \left[ \frac{2\sqrt{\pi}\gamma\Theta}{\mu\alpha} e^{-\alpha_{f}L}(1 - e^{-\alpha_{f}L}) \right] \qquad (B.3)$$

$$\Theta = \frac{T \cdot \nu_{p} \cdot \Delta \nu_{B}}{k_{polar} \cdot 2 \cdot \nu_{B}} \qquad (B.4)$$

$$\gamma = \frac{g_{B} k_{polar}}{A_{eff}} \qquad (B.5)$$

In 1972, Smith et al.[187] approximated the value of $g_{th}$ is 21 in silica fiber with $\mu = 1$ and large fiber losses coefficient. This value is over-estimated due to the available fibers in 1972 [33]. $\mu = 1$ means that the input pump and the Stokes power are the same order. In the manuscript $\mu = .01$ because it enables to be at the middle of the transition between the spontaneous ans stimulated regime. This ratio means that the Brillouin threshold is reached when the back-scattered power is equal to 1% of the pump power [83, 84]. We estimate $g_{th}$ equal to 11.24 for 26 km SMF-28, considering a Brillouin gain coefficient extracted using the self-Heterodyne method and material parameters (see table 2.3,2.2).
<figure><img src="image_72.png" /></figure>
# Appendix C

## Gain distribution

Then the expression of the gain for a stimulated Brillouin regime is Gaussian shape:

$$g_B(\nu) = \exp \left[ - \ln(2) \left( 2 \frac{\nu}{\Delta\nu_B} \right)^2 \right] \quad (C.1)$$

for

$$\nu = \pm \Delta\nu_B/2 \quad (C.2)$$

$$g = g_B \exp(-\ln(2)) = g_B/2 \quad (C.3)$$

If the ratio between modes under the Brillouin gain is 20 dB, then the corresponding frequency is written:

$$g_B = \exp \left[ - \ln(2) \left( 2 \frac{\nu}{\Delta\nu_B} \right)^2 \right] = \frac{g_B}{100} \quad (C.4)$$

So that

$$\exp \left[ - \ln(2) \left( 2 \frac{\nu_{\pm}}{\Delta\nu_B} \right)^2 \right] = \exp(4605) \quad (C.5)$$

And

$$\exp \left[ - \ln(2) \left( 2 \frac{\nu_{\pm}}{\Delta\nu_B} \right)^2 \right] = \exp(-4605) \quad (C.6)$$

Finally,

$$\nu_{\pm} = \pm \frac{1}{2} \sqrt{\frac{4605}{\ln(2)}} \Delta\nu_B = \pm 1289 \Delta\nu_B \quad (C.7)$$

To consider a side-mode suppression (SMSR) of 20 dB, the FSR has to be greater than 2.58 $\Delta\nu_B$ (see figure(C.1)). In practice, as the gain cannot be considered as homogeneous, it is difficult to fix a ratio between FSR and the gain bandwidth in order to assure a single-mode operation. It turns out that a ratio of $\Delta\nu_{FSR}/\Delta\nu_B$ is a good indication to have a single mode operation even if there could be situation where it is not a sufficient one. Then we will consider in the following the ratio $\Delta\nu_{FSR}/\Delta\nu_B$ as a reference.
<figure><img src="image_73.png" /><figcaption>Figure C.1: Representation of the Brillouin gain and side mode suppression</figcaption></figure>
# Appendix D

## Brillouin shift and gain detuning in multi-Stokes Brillouin laser

Gain detuning is inherent in the multi-Stokes Brillouin lasers. The Brillouin laser is realized by the matching of the $N \times \Delta\nu_{FSR}$ of the cavity and Brillouin shift $\nu_B$. It depends on the refractive index, acoustic wave velocity of the medium and pump wavelength.

$$ \nu_B = \frac{2nV_A}{\lambda_P} \qquad (D.1) $$

The Brillouin shift varies with pump wavelength by ratio of 7 MHz/nm. As we know that the each generated Stokes waves are downshift from the pump frequency, so the Brillouin shift will be change from Stokes 1 to Stokes 2. Figure (D.1) gives a picture of

<figure><img src="image_74.png" /><figcaption>Figure D.1: Illustration of gain detuning in multi-Stokes Brillouin laser</figcaption></figure>

detuning in multi-Stokes Brillouin laser. The pump laser frequency $\nu_P$ (red) is in red while resonant modes are in gray color. Shared region is the Brillouin gain line, sharp
peaks are the Stokes orders. Here we assume that ideal conditions like constant pressure and temperature so that $V_A$ remains constant.

The Brillouin shift $\nu_{B1}$ due to the pump ($\nu_p = c/\lambda_p$) may be written as:

$$ \nu_{B1} = Z_A \nu_p \qquad (D.2) $$

with $Z_A = \frac{2nV_A}{c}$.

Then the frequency $\nu_{maxgS1}$ of the gain maximum for Stokes 1 is at:

$$ \nu_{maxgS1} = \nu_p - \nu_{B1} = \nu_p - Z_A \nu_p \qquad (D.3) $$

And the Stokes frequency $\nu_{S1}$ is given by:

$$ \nu_{S1} = \nu_p - N \Delta \nu_{FSR} \qquad (D.4) $$

The detuning $\Delta\nu_{S1}$ is for the first order Stokes:

$$ \Delta\nu_{S1} = \nu_{S1} - \nu_{maxgS1} = Z_A \nu_p - N \Delta\nu_{FSR} \qquad (D.5) $$

N is the number of free spectral range necessary to have a resonance under the gain curve.

When the Stokes 2 is generated from the first order Stokes, the associated Brillouin shift is slightly different from $\nu_{B1}$ and is written:

$$ \nu_{B2} = Z_A \nu_{S1} = Z_A \nu_p - Z_A N \Delta \nu_{FSR} \qquad (D.6) $$

The frequency of the maximum gain is then:

$$ \nu_{maxgS2} = \nu_{maxgS1} - \nu_{B2}\nu_{maxgS2} = \nu_p - 2Z_A\nu_p + Z_A N \Delta\nu_{FSR} \qquad (D.7) $$

And the Stokes frequency is written as:

$$ \nu_{S2} = \nu_p - 2N \Delta \nu_{FSR} \qquad (D.8) $$

The detuning $\Delta\nu_{S2}$ is for the second order Stokes:

$$ \Delta\nu_{S2} = \nu_{S2} - \nu_{maxgS2} = 2Z_A\nu_p - Z_A N \Delta\nu_{FSR} - 2N \Delta\nu_{FSR} \qquad (D.9) $$

$$ \Delta\nu_{S2} = \nu_{S2} - \nu_{maxgS2} = 2Z_A\nu_p - (Z_A + 2)N \Delta\nu_{FSR} \qquad (D.10) $$

For Stokes 3,

$$ \nu_{B3} = Z_A \nu_{S2} = Z_A \nu_p - Z_A 2N \Delta \nu_{FSR} \qquad (D.11) $$
$$ \nu_{maxgS3} = \nu_{maxgS2} - \nu_{B3} \qquad (D.12) $$

$$ \nu_{maxgS3} = \nu_p - 3Z_A\nu_p + Z_A N \Delta\nu_{FSR} + Z_A 2N \Delta\nu_{FSR} \qquad (D.13) $$

$$ \nu_{maxgS3} = \nu_p - 3Z_A\nu_p + 3Z_A N \Delta\nu_{FSR} \qquad (D.14) $$

$$ \nu_{S3} = \nu_p - 3N \Delta\nu_{FSR} \qquad (D.15) $$

$$ \Delta\nu_{S3} = \nu_{S3} - \nu_{maxgS3} = 3Z_A\nu_p - Z_A 3N \Delta\nu_{FSR} - 3N \Delta\nu_{FSR} \qquad (D.16) $$

$$ \Delta\nu_{S3} = \nu_{S3} - \nu_{maxgS3} = 3Z_A\nu_p - 3(Z_A - 1)\Delta\nu_{FSR} \qquad (D.17) $$

The general expressions for the order $\zeta$ are then given by:

For the Stokes frequency

$$ \nu_{S\zeta} = \nu_p - \zeta N \Delta\nu_{FSR} \qquad (D.18) $$

For the Brillouin shift

$$ \nu_{B\zeta} = Z_A \nu_{S(\zeta-1)} = Z_A \nu_p - Z_A (\zeta - 1) N \Delta \nu_{FSR} \qquad (D.19) $$

For the frequency of the gain maximum

$$ \nu_{maxgS\zeta} - \nu_{maxgS(\zeta-1)} = -\nu_{B\zeta} \qquad (D.20) $$

or

$$ \nu_{maxgS\zeta} - \nu_{maxgS(\zeta-1)} = -Z_A \nu_p + (\zeta - 1) Z_A N \Delta \nu_{FSR} \qquad (D.21) $$

With

$$ \nu_{maxgS0} = \nu_p \qquad (D.22) $$

Which gives from series law

$$ \nu_{maxgS\zeta} = \nu_p - \zeta Z_A \nu_p + \frac{1}{2} \zeta (\zeta - 1) Z_A N \Delta \nu_{FSR} \qquad (D.23) $$

It follows that the detuning is equal to:

$$ \Delta\nu_{S\zeta} = \nu_{S\zeta} - \nu_{maxgS\zeta} = \zeta Z_A \nu_p - \frac{1}{2} \zeta (\zeta - 1) Z_A N \Delta\nu_{FSR} - \zeta N \Delta\nu_{FSR} \qquad (D.24) $$
When the first detuning is null, then $Z_A\nu_p = N\Delta\nu_{FSR}$ and the detuning is then written as:

$$ \Delta\nu_{S\zeta} = \nu_{S\zeta} - \nu_{\text{maxgS\zeta}} = \zeta Z_A \nu_p - \frac{1}{2}\zeta(\zeta - 1)Z_A^2 \nu_p - \zeta Z_A \nu_p \quad (\text{D.25}) $$

It follows the general formula for the detuning

$$ \Delta\nu_{S\zeta} = -\frac{1}{2}\zeta(\zeta - 1)Z_A^2\nu_p \quad (\text{D.26}) $$

<table><thead><tr><th>Variable</th><th>Expression</th></tr></thead><tbody><tr><td>Stokes frequency</td><td>ν<sub>Sζ</sub> = ν<sub>p</sub> - ζNΔν<sub>FSR</sub></td></tr><tr><td>Brillouin Shift</td><td>ν<sub>Bζ</sub> = Z<sub>A</sub>ν<sub>p</sub> - Z<sub>A</sub>(ζ - 1)NΔν<sub>FSR</sub></td></tr><tr><td>Frequency of the maximum gain</td><td>ν<sub>maxgSζ</sub> = ν<sub>p</sub> - ζZ<sub>A</sub>ν<sub>p</sub> + <sup>1</sup>&frasl;<sub>2</sub>ζ(ζ - 1)Z<sub>A</sub>NΔν<sub>FSR</sub></td></tr><tr><td>Detuning</td><td>Δν<sub>Sζ</sub> = ζZ<sub>A</sub>ν<sub>p</sub> - <sup>1</sup>&frasl;<sub>2</sub>ζ(ζ - 1)Z<sub>A</sub>NΔν<sub>FSR</sub> - ζNΔν<sub>FSR</sub></td></tr><tr><td>Detuning if S1 detuning is 0</td><td>Δν<sub>Sζ</sub> = -<sup>1</sup>&frasl;<sub>2</sub>ζ(ζ - 1)Z<sub>A</sub><sup>2</sup>ν<sub>p</sub></td></tr></tbody></table>

From these equations, we can obtain Stokes detuning for any stokes orders. In table (D.1) gives the values of detuning, wavelength and the Brillouin shift for Stokes order up to 5. These detuning are in the MHz range. It has thus an important influence on the noise properties. For Stokes 2, the detuning is 610 kHz. The detuning is increasing with number of Stokes orders.

<table><caption>Table D.1: Gain detuning in cascaded Brillouin laser</caption><thead><tr><th>Order</th><th>wavelength (nm)</th><th>ν<sub>Bζ</sub> (MHz)</th><th>Δν<sub>Sζ</sub> (MHz)</th></tr></thead><tbody><tr><td>S1</td><td>1550.09</td><td>10869.93</td><td>0</td></tr><tr><td>S2</td><td>1550.17</td><td>10869.32</td><td>.61</td></tr><tr><td>S3</td><td>1550.26</td><td>10868.71</td><td>1.83</td></tr><tr><td>S4</td><td>1550.35</td><td>10868.10</td><td>3.66</td></tr><tr><td>S5</td><td>1550.44</td><td>10867.49</td><td>6.10</td></tr></tbody></table>
# Appendix E

## List of Publications

### Journal papers

* **Ananthu Sebastian**, Stéphane Trebaol, Pascal Besnard; *Intracavity Brillouin gain characterization based on cavity ringdown spectroscopy* OSA Continuum, OSA Publishing, 2019, 2 (12), pp.3539-3545.

* **Ananthu Sebastian**, Irina V. Balakireva, Schadrac Fresnel, Stéphane Trebaol, and Pascal Besnard; *Relative intensity noise in a multi-Stokes Brillouin laser* Optics Express, Optical Society of America, 2018, 26 (26), pp.33700-33711.

### Conferences contributions

* **Ananthu Sebastian**, Irina V Balakireva, Stéphane Trebaol, Pascal Besnard; *Relative intensity noise in a multi-Stokes Brillouin laser*, Workshop on OptoMechanics and Brillouin Scattering (WOMBAT). March 2019, Tel-Aviv, Israel.

* **Ananthu Sebastian**, Stéphane Trebaol, Pascal Besnard; *Intracavity Brillouin Gain Characterization*, International Topical Meeting on Microwave Photonics (MWP), Oct 2018, Toulouse, France

* **Ananthu Sebastian**, Schadrac Fresnel, Stéphane Trebaol, Frederic Ginovart, Pascal Besnard; *Improvement of noise reduction in fiber Brillouin lasers due to multi-Stokes operation*, SPIE Photonics Europe 2018, Apr 2018, Strasbourg, France.
* Frederic Ginovart, Schadrac Fresnel, **Ananthu Sebastian**, Stéphane Trebaol, Sophie La Rochelle, Pascal Besnard; *Phase-Variance reduction of high order Stokes lines in Brillouin fiber lasers*, Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference 2017 (CLEO-Europe/EQEC 2017), Jun 2017, Munich, Germany.

* Schadrac Fresnel, **Ananthu Sebastian**, Stéphane Trebaol, Christelle Pareige, Frederic Ginovart, Pascal Besnard, Sophie La Rochelle; *Laser Brillouin Multi-Stokes pour la réduction du bruit de fréquence et d'intensité*. JNOG 2017, Limoges France.
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<figure><img src="image_75.png" /></figure>
Dynamique de bruit dans les lasers
Brillouin multi-Stokes

La diffusion Brillouin stimulée (SBS) est un processus d'interaction cohérent, pour lequel
la lumière est diffusée à partir des ondes acoustiques générées optiquement. C'est un
outil puissant pour le traitement des micro-ondes et des signaux optiques, la détection
distribuée spatialement et la spectroscopie. Les lasers Brillouin suscitent un très grand
intérêt pour leur capacité à produire des largeurs de raie très fines (de l'ordre du Hz-kHz
; lasers ultra-cohérents).

Cette thèse est consacrée à la compréhension des lasers à fibre Brillouin en anneau, fonctionnant avec plusieurs ordres de Stokes. Mon travail a été réalisé dans le cadre du projet SOLBO (SOurces Laser Brillouin cOhérent à faible bruit) de l'appel FUI (Fonds Unique Interministériel AAP20) avec l'aide de BPI France, le cluster "Images et réseaux" et en collaboration avec IDIL Fibre Optiques (chef de projet), Lumibird, THALES, ISCR (Institut des Sciences Chimiques de Rennes), Photonics Bretagne.

L'objectif principal était de concevoir un laser à fibre Brillouin à faible coût, avec une largeur de raie intégrée inférieure au kHz, par IDIL. Pour l'Institut Foton, l'objectif était d'explorer la physique des lasers à fibre multi-Stokes Brillouin et d'essayer de connaître les limites fondamentales la réduction du bruit de fréquence pour ces sources, que ce soit pour des cavités avec un seul ordre Stokes simples ou avec plusieurs ordres Stokes. Dans cette thèse, je propose:

* une nouvelle technique de mesure de gain Brillouin, particulièrement adaptée aux cavités courtes et ne nécessitant pas de fortes puissances de pompage optique (quelques dizaines de milliWatts),

• une comparaison du fonctionnement du laser entre le pompage résonnant et non
résonnant,

* une modélisation de nos résultats expérimentaux par une approche analytique, permettant d'expliquer les mécanismes physiques de ces lasers,

* une discussion sur les limitations fondamentales de ces appareils et leurs avantages.

Le premier chapitre de ce manuscrit présente des généralités sur le bruit laser. Le bruit d'intensité relatif (RIN : Relative Intensity Noise) des lasers est théoriquement expliqué et mesuré expérimentalement à l'aide d'un laser DFB commercial (KOHERAS). De plus, le bruit de fréquence des lasers est expliqué et lié à la largeur de raie du laser. Une illustration est donnée grâce à un laser accordable (Tunics OM) et la largeur de ligne est mesurée en utilisant la technique dite *beta-line*, qui est comparée à une technique auto-hétérodyne. Une caractérisation importante de ces qualifications de bruit est l'évaluation du bruit de fond du banc et son analyse. Cette analyse a été réalisée en utilisant différentes longueurs de retard dans l'interféromètre, qui est utilisé comme convertisseur d'intensité pour les fluctuations de fréquence. Cela a été discuté en détail. De plus, afin d'obtenir une stabilité à long terme (de l'ordre de quelques heures) et afin de faciliter nos mesures, nous avons démontré la stabilisation d'un laser sur un résonateur à fibre optique en utilisant la technique Pound-Drever-Hall (PDH). Cette technique nous a permis de réduire la largeur intégrée de raie du laser de 203 kHz à 2 kHz.

Dans le deuxième chapitre, nous avons présenté une introduction générale des diffusions de Brillouin spontanées et stimulées. Nous avons discuté et mesuré les paramètres clés des diffusions de Brillouin, qui sont la largeur de bande de gain, le décalage de fréquence et le coefficient de gain de Brillouin (gain maximum). Une nouvelle méthode de caractérisation du gain de Brillouin basée sur la méthode de *cavity ring down* (CRDM) a été introduite. La caractérisation de la cavité à l'aide du CRDM, permet d'identifier le régime de couplage et les pertes intracavités du résonateur, soient le taux de couplage entre le coupleur et la cavité ainsi que les pertes linéiques intra-cavité. Ces mesures sont importantes car elles donnent directement accès aux paramètres utilisés dans notre modèle théorique. Enfin, nous avons montré que le coefficient de gain de Brillouin estimé à l'aide du CRDM est en bon accord avec les résultats obtenus par des méthodes standard (nous avons vérifié avec la méthode auto-hétérodyne). La méthode CRDM offre la possibilité
bilité d'obtenir les paramètres de gain matériau (il s'agit du paramètre de gain matériau effectif correspondant aux modes acoustiques des fibres standard (SMF28)) et de cavité, en une seule expérience, en utilisant des puissances de pompe optique de l'ordre de quelques à dizaines de milliWatts, pour des cavités de faible longueur (mètre, millimètre, micromètre). C'est un intérêt significatif par rapport à d'autres méthodes.

Dans le troisième chapitre, nous avons passé en revue le développement du laser Brillouin au cours des dernières décennies. Cela va du laser en anneau à fibre aux lasers Brillouin intégrés à puce. Après avoir expliqué la différence entre un laser Brillouin sous pompage résonant et non résonant, le caractère, le caractère monomode ou multimode du fonctionnement du laser Brillouin est expliqué. Les sauts de modes (longitudinaux de la cavité) du laser Brillouin non résonant sont illustrés à l'aide d'une cavité Brillouin de 20 mètres. Nous montrons la suppression de ces sauts de mode à l'aide d'une technique de verrouillage PDH. Cela nous a permis d'obtenir un laser Brillouin non résonnant avec une stabilité à long terme de plusieurs heures. De plus, nous sommes en mesure de contrôler des paramètres clés tels que les facteurs de cavité (facteur Q, pertes de couplage, pertes internes), le désaccord entre la résonance de Stokes laser et le gain maximum (ou de manière équivalente le décalage Brillouin), le bruit de la pompe. Toutes ces améliorations nous ont permis d'avoir un contrôle fin du laser, et de faire une comparaison qualitative et quantitative entre les modèles et les résulats expérimentaux. L'analyse de ce troisième chapitre est réalisée à l'aide d'un modèle standard de transfert de bruit pour les lasers Brillouin non résonants. Nous avons démontré que notre système peut atteindre une largeur de raie intrinsèque de 1,3 Hz avec une réduction de 37 ±1 dB par rapport à la largeur de raie du laser de la pompe. De plus, nous avons également montré que la largeur de raie intrinsèque du laser Brillouin peut atteindre le niveau de mHz en cascadant deux lasers Brillouin non résonants.

Dans le dernier chapitre, la dynamique du bruit du laser Brillouin résonnant est étudiée expérimentalement à l'aide d'une cavité en anneau de fibre. Un modèle généralisé aux lasers Brillouin multi-Stokes a été développé à l'Institut Foton; Nous avons montré que nos mesures expérimentales sont en bon accord avec les simulations numériques pour le laser Brillouin. Le RIN des lasers Brillouin à plusieurs ordres multi-Stokes a ensuite été étudié expérimentalement et numériquement en termes de puissance de pompage, de durée de vie des photons et de désaccord de gain. Nous avons confirmé que dans le régime que nous appelons en cascade et correspondant au fonctionnement avec plusieurs ordres
de Stokes, et pour certaines gammes de fréquences, le RIN peut être réduit par rapport aux régimes monomodes de façon très significative (~ 40 dB). Cette réduction est due à la saturation de la puissance de la pompe intracavité due au processus laser.

La durée de vie de la cavité photonique change principalement la fréquence de relaxation du laser Brillouin multi-Stokes.

Le désaccord entre la résonance de Stokes laser et le maximum de gain de Brillouin a une grande influence sur la réponse du RIN, même pour de petites valeurs de désaccord (~ environ 100 kHz). Nous détaillons théoriquement et expérimentalement l'augmentation de RIN par l'influence du désaccord de gain.

Enfin, nous avons tenté d'explorer les limites de la réduction du bruit de fréquence à l'aide de deux lasers Brillouin. Nous avons montré qu'il est possible de mettre en cascade la réduction fréquence-bruit en cascadant deux lasers Brillouin. Cependant notre plancher de bruit de banc est rapidement atteint et nous a empêché d'explorer expérimentalement toute limitation de réduction du bruit même si le bruit thermique sera en tout cas une limitation fondamentale. Néanmoins, une largeur de ligne intrinsèque de l'ordre de 100 mHz est facile à obtenir dans une configuration mono-Stokes ou dans une configuration multi-Stokes.

D'autres techniques sont actuellement explorées pour surmonter les limites de bruit de fond de nos bancs de mesure de bruit de fréquence afin de nous permettre d'atteindre les limites fondamentales de réduction du bruit pour la largeur de raie intrinsèque. Il va de soi que le bruit basse fréquence (bruit de scintillement, marche aléatoire) associé aux fluctuations thermiques et mécaniques nécessite un asservissement sur une référence métrologique pour obtenir de petites largeurs de raie intégrées. Nous avons réussi à atteindre la largeur de raie intégrée en kHz en utilisant, comme référence, la fréquence de résonance d'un interféromètre à fibres.

Cependant, nous avons souligné qu'il existe des limitations fondamentales pour les lasers Brillouins avec plusieurs ordres de Stokes. Nous avons démontré expérimentalement, en accord avec des prédictions théoriques montrant la pénalité causée par les ondes anti-Stokes, qu'une dégradation du bruit de fréquence est observée. Cela peut être abaissé en opérant loin du seuil de l'ordre de Stokes analysé.

De plus, le décalage de Brillouin dépend de l'ordre de Stokes car son décalage de fréquence dépend de la longueur d'onde. Cela générera un désaccord inhérent lorsque l'ordre de Stokes est augmenté. Comme le montre précisément cette thèse, ce désaccord
dégrade le RIN et le bruit de fréquence par couplage amplitude-phase au sein du processus non linéaire. Il s'avère que c'est une limitation sévère pour les lasers Brillouin avec plusieurs ordres de Stokes, car un petit désaccord (moins de 1 MHz) affectera à la fois le RIN et le bruit de fréquence.

Nous soulignons le fait que de nombreuses caractéristiques du bruit de fréquence sont liées aux propriétés RIN par un couplage entre intensité et phase.

Nous montrons que le processus en cascade modifie la dynamique du laser Brillouin par rapport à celles d'un laser Brillouin monomode avec une seule composante de Stokes du premier ordre. Nos résultats expérimentaux sont en excellent accord avec nos simulations numériques, obtenues grâce à notre système non linéaire décrivant le fonctionnement d'un laser Brillouin multi-Stokes. Ce bon accord est principalement dû à notre capacité à :

* obtenir des valeurs très précises des paramètres de la cavité et du coefficient de gain Brillouin en utilisant la technique CRDM ;

* atteindre une stabilité à long terme (plusieurs dizaines d'heures) ;

* contrôler finement le désaccord entre la résonance de Stokes du laser et la fréquence du maximum de gain Brillouin.

Nous démontrons expérimentalement pour la première fois que le bruit de fréquence est dégradé en présence de diffusion anti-Stokes Brillouin.

Nous montrons également qu'un désaccord de gain de l'ordre de quelques centaines de kHz peut dégrader la réduction du bruit d'intensité ou encore augmenter la largeur de raie par couplage amplitude-phase.

Toutes ces observations très fines permettent ainsi de fixer les limites fondamentales de tels systèmes laser telles que : l'augmentation du bruit due aux ordres anti-Stokes ; le rôle du bruit de la pompe et son éventuelle corrélation avec la finesse de la cavité ; l'effet du désaccord inhérent aux ordres de Stokes supérieurs.

Toutes ces conclusions sont essentielles à la conception et à l'ingénierie de ces lasers à fibre Brillouin, qui suscitent actuellement beaucoup d'intérêt comme en témoignent les travaux en cours dans la communauté scientifique.

Cette thèse contribue à une meilleure compréhension des lasers Brillouin multi-Stokes.