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# Noise dynamics in multi-Stokes Brillouin laser

Ananthu Sebastian

► To cite this version:

Ananthu Sebastian. Noise dynamics in multi-Stokes Brillouin laser. Optics / Photonic. Université de Rennes 1, 2020. English. NNT: tel-03215224

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Submitted on 3 May 2021

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# THESE DE DOCTORAT DE

L'UNIVERSITE DE RENNES 1

ECOLE DOCTORALE N° 596

Matière, Molécules, Matériaux

Spécialité : « Photonique »

Par

Ananthu SEBASTIAN

## Noise dynamics in multi-Stokes Brillouin laser

Thèse présentée et soutenue à Lannion, le 18 décembre 2020

Unité de recherche : INSTITUT FOTON (UMR 6082/CNRS)

## Rapporteurs avant soutenance :

Yves JAOUEN Professeur, Telecom Paris Tech

Olivier LLOPIS Directeur de recherche CNRS, LAAS-CNRS

## Composition du Jury :

Président : Medhi ALOUINI

Professeur, Université de Rennes 1/ Institut Foton

Examineurs : Anne AMY-KLEIN

Professeure, Université Sorbonne Paris Nord/ LPL

Jean-Charles BEUGNOT Chargé de recherche, Institut Femto-ST

Carlos ALONSO-RAMOS Chargé de recherche CNRS, C2N

Dir. de thèse : Pascal BESNARD

Professeur, ENSSAT( Université de Rennes 1)/ Institut Foton

Co-dir. de thèse : Stéphane TREBAOL

Maitre de conférences, ENSSAT( Université de Rennes 1)/ Institut Foton
<figure><img src="image_2.png" /></figure>
Acknowledgement

I would like to extend my sincere gratitude towards the remarkable individuals who have supported and helped me throughout my PhD at Institut FOTON. It has been a time of intense learning for me, not only in the scientific field but also on a personal level.

First and foremost, I wish to thank my principal supervisor Pascal BESNARD for
giving me the opportunity to pursue my PhD at the University of Rennes 1, France. His
excellent guidance, confidence, supervision and continuous encouragement have been a
constant inspiration that made so many things possible throughout my research. Pascal
always trusted me and find time to discuss during heavy administrative tasks incumbent
on him. In addition, I would like to thank my co-supervisor, Stephane TREBAOL for
his guidance, friendship, and extended support in the course of my research. All my
supervisor's motivation and immense knowledge always helped me during my PhD and
writing this thesis.

Besides supervisors, I would like to acknowledge my jury members Yves JAOUEN and Olivier LLOPIS for accepting to be the reporter of this manuscript. Many thanks for their very relevant remarks. I thank Anne AMY-KLEIN from LPL, Jean-Charles BEUGNOT from Institut Femto-ST, Carlos ALONSO-RAMOS from C2N and Medhi ALOUINI from Institut Foton for their constructive questions during the defense.

With a special mention to Irina V Balakireva, who contributed the numerical simula-
tions and theoretical model for noise analysis. I would like to acknowledge the valuable
inputs from Schadrac Fresnel and Mohamed Omar Sahni, who contributed to many fruit-
ful discussions and showed me all the details of experimental physics in the initial phase
of my PhD, which always helped me to achieve valuable insights into Brillouin laser.

I would also like to thank all of the academic and support staff of the Institut FOTON and ENSSAT for their fruitful discussions and assistance throughout my research period.
I would also thank all my colleagues and friends, Antoine Congar, Dominique Mammez,
Nessim Jebali, Joseph Slim, Georges Perin, Louis Ruel, Rasool Saleem and Amith Karu-
vath, who made my life simple, enjoyable and memorable at Institut FOTON. It was a
great pleasure to work with you and thank you so much for your friendship and for all
the good times!

Finally, I could not conclude these thanks without having thought of my inspiring
parents and sister, fantastically supportive family members, and my enchanting friends.

Thank you

Ananthu SEBASTIAN
Abstract

Stimulated Brillouin Scattering (SBS) is a coherent interaction process in which light is scattered from optically generated acoustic waves. It is a powerful tool for microwave and optical signal processing, distributed sensing and spectroscopy. Brillouin lasers are attracting a lot of interest for their ability to produce ultra coherent linewidths.

This thesis is devoted to the understanding of noise properties of Brillouin fiber ring lasers, operating with multiple Stokes orders. First, we present a technique based on the cavity ringdown method, which allows to characterize the Brillouin gain coefficient directly from probing the laser cavity. Its advantages are to obtain parameters from a single experiment with low optical powers (some 10 milliwatts) for short cavities (a few meters long, or integrated cavities).

Secondly, it is shown that an intrinsic linewidth of a few tens of mHz can be easily obtained by cascading two non-resonant Brillouin lasers (for which the pump performs a single pass inside the cavity). In order to obtain these results, the long-term stability has been improved by using a Pound-Drever-Hall servo loop, which allows us to compare our analytical and experimental results. Unfortunately, we were unable to explore the fundamental limits of noise reduction due to the noise floor of our bench.

Thirdly, one of the major works of this thesis is the theoretical and experimental study of the noise properties, including frequency noise and relative intensity noise, of a resonant Brillouin laser (for which pump and Stokes waves are resonant inside the cavity). In particular, the impacts of the fiber-ring-cavity quality factor, Brillouin gain detuning, are evaluated very precisely on the laser RIN features such as amplitude noise reduction and relaxation frequency.

We emphasize the fact that many characteristics of the frequency noise are related to the RIN properties by a coupling between intensity and phase. We show that the cascade process modifies the dynamics of the Brillouin laser when compared to those of a
single-mode Brillouin laser with a single first-order Stokes component. Our experimental results are in excellent agreement with our numerical simulations, obtained thanks to our non-linear system describing the operation of a multi-Stokes Brillouin laser. This good match is mainly due to our ability:

* to obtain very precise values of the cavity parameters and the Brillouin gain coefficient using the CRDM technique;

* to achieve long-term stability (hours);

* to finely control the detuning between the laser Stokes resonance and the frequency of the Brillouin gain maximum.

We demonstrate experimentally for the first time that frequency noise is degraded in the presence of anti-Stokes Brillouin scattering. We also show that a gain detuning of the order of a few hundred kHz can degrade the intensity noise reduction or also increase the linewidth by amplitude-phase coupling. All these very fine observations thus allow us to set the fundamental limits of such laser systems such as: the increase in noise due to anti-Stokes orders; the role of pump noise and its possible interrelation with cavity finesse; the effect of the detuning inherent to higher Stokes orders.

All these conclusions are key to the design and engineering of these Brillouin fiber lasers, which are currently attracting a great deal of interest as evidenced by the work in progress in the scientific community. This PhD thesis contributes to a better understanding of multi-Stokes Brillouin lasers.

**Keywords** : Nonlinear optics, Stimulated Brillouin Scattering, Brillouin laser, frequency noise, relative intensity noise, laser linewidth, optical resonator, cavity ring down spectroscopy, frequency stabilization.
Résumé

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 et la spectroscopie. Les lasers Brillouin suscitent un très grand intérêt pour
leur capacité à produire des largeurs de raie ultra cohérentes.

Cette thèse est consacrée à la compréhension des propriétés de bruit des lasers à fibre
Brillouin en anneau, fonctionnant avec de multiples ordres de Stokes. Tout d'abord, nous
présentons une technique basée sur la méthode de ringdown de la cavité, qui permet de
caractériser le coefficient de gain Brillouin directement à partir du sondage de la cavité
laser. Ses avantages sont d'obtenir des paramètres à partir d'une seule expérience avec
de faibles puissances optiques (quelques 10 milliwatts) pour des cavités courtes (quelques
mètres de long, ou cavités intégrées).

Deuxièmement, il est démontré qu'une largeur de raie intrinsèque de quelques dizaines
de mHz peut être facilement obtenue en cascadant deux lasers Brillouin non résonants
(pour lesquels la pompe effectue un seul passage à l'intérieur de la cavité). Afin d'obtenir
ces résultats, la stabilité à long terme a été améliorée en utilisant une boucle d'asservissement
de type Pound Drever Hall, ce qui nous permet de comparer nos résultats analytiques et
expérimentaux. Malheureusement, nous n'avons pas exploré les limites fondamentales de
la réduction du bruit en raison du plancher de bruit de notre banc de mesure.

Troisièmement, un des travaux majeurs de cette thèse est l'étude analytique et ex-
périmentale des propriétés du bruit, y compris le bruit de fréquence et le bruit relatif
d'intensité, d'un laser Brillouin résonant (pour lequel, les ondes de pompe et de Stokes
sont résonantes à l'intérieur de la cavité). En particulier, les impacts du facteur de qualité
de la cavité fibrée en anneau, le désaccord de gain Brillouin ont été évalués très précisé-
ment sur les caractéristiques du RIN du laser telles que la réduction de l'amplitude du
bruit et la fréquence de relaxation.

Nous soulignons le fait que de nombreuses caractéristiques du bruit de fréquence sont liées aux propriétés du RIN par un couplage entre l'intensité et la phase. Nous montrons que le processus en cascade modifie la dynamique du laser Brillouin par rapport à celle d'un laser Brillouin monomode avec une seule composante de Stokes de premier ordre. Nos résultats expérimentaux sont en excellent accord avec nos simulations, obtenues grâce à notre système non linéaire décrivant le fonctionnement d'un laser Brillouin multi-Stokes. Cette bonne concordance est principalement due 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 ;

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

* à 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 d'une diffusion Brillouin anti-Stokes. 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 également augmenter la largeur de raie par un couplage amplitude-phase. Toutes ces observations très fines nous permettent donc de fixer les limites fondamentales de tels systèmes laser comme : l'augmentation du bruit due aux ordres anti-Stokes ; le rôle du bruit de pompe et son interrelation possible avec la finesse de la cavité ; l'effet du désaccord inhérent aux ordres de Stokes plus élevés.

Toutes ces conclusions sont les clés de la conception et de 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 de doctorat contribue à une meilleure compréhension des lasers Brillouin multi-Stokes.

**Mots clefs** : Optique non linéaire, diffusion Brillouin stimulée, laser Brillouin, bruit de fréquence, bruit relatif d'intensité, largeur de raie laser, résonateur optique, spectroscopie cavity ring-down, stabilisation de fréquence.
# Contents

<table><tr><td>Contents</td><td>ix</td></tr><tr><td>List of Figures</td><td>xii</td></tr><tr><td>List of Tables</td><td>xvii</td></tr><tr><td>1 Noise in lasers</td><td>7</td></tr><tr><td>1.1 Introduction</td><td>7</td></tr><tr><td>1.2 Origin of noise in lasers</td><td>7</td></tr><tr><td>1.3 Relative intensity noise (RIN)</td><td>9</td></tr><tr><td>1.3.1 Definition of relative intensity noise</td><td>9</td></tr><tr><td>1.3.2 RIN measurement method</td><td>10</td></tr><tr><td>1.3.3 Determination of thermal noise</td><td>10</td></tr><tr><td>1.3.4 Determination of shot noise</td><td>11</td></tr><tr><td>1.3.5 Determination of relative intensity noise</td><td>12</td></tr><tr><td>1.4 Phase noise</td><td>13</td></tr><tr><td>1.4.1 Frequency noise in laser</td><td>14</td></tr><tr><td>1.4.2 Relation between the power spectral density of frequency noise and optical spectrum</td><td>15</td></tr><tr><td>1.4.3 Experimental characterization of frequency noise</td><td>19</td></tr><tr><td>1.4.4 Noise floor of the measurement bench</td><td>21</td></tr><tr><td>1.4.5 Example of noise measurement</td><td>23</td></tr><tr><td>1.5 Laser-frequency stabilization</td><td>24</td></tr><tr><td>1.5.1 Top-of-fringe locking</td><td>27</td></tr><tr><td>1.5.2 Pound-Drever-Hall technique</td><td>29</td></tr><tr><td>1.6 Conclusion</td><td>31</td></tr><tr><td>2 Brillouin gain characterization using cavity ring-down spectroscopy</td><td>33</td></tr><tr><td>2.1 Introduction</td><td>33</td></tr><tr><td>2.2 Generalities on stimulated Brillouin scattering</td><td>33</td></tr><tr><td>2.2.1 Spontaneous Brillouin scattering</td><td>34</td></tr><tr><td>2.2.2 Stimulated Brillouin scattering</td><td>36</td></tr><tr><td>2.2.3 Brillouin gain</td><td>38</td></tr><tr><td>2.2.4 Brillouin threshold</td><td>40</td></tr><tr><td>2.3 Experimental characterization of Brillouin gain</td><td>42</td></tr><tr><td>2.3.1 Self-heterodyne technique</td><td>43</td></tr><tr><td>2.3.2 SBS threshold</td><td>46</td></tr><tr><td>2.4 Cavity ring-down method</td><td>48</td></tr><tr><td>2.4.1 Cavity parameters description</td><td>50</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>2.4.2</td>
      <td>Stationary approach:</td>
      <td>51</td>
    </tr>
    <tr>
      <td>2.4.2.1</td>
      <td>Transmission and coupling regime</td>
      <td>52</td>
    </tr>
    <tr>
      <td>2.4.2.2</td>
      <td>Q-factor and coupling coefficients</td>
      <td>53</td>
    </tr>
    <tr>
      <td>2.4.3</td>
      <td>Dynamical approach</td>
      <td>54</td>
    </tr>
    <tr>
      <td>2.4.4</td>
      <td>Estimation of Brillouin gain coefficient</td>
      <td>56</td>
    </tr>
    <tr>
      <td>2.4.5</td>
      <td>Experimental setup for Brillouin gain measurement using CRDM</td>
      <td>58</td>
    </tr>
    <tr>
      <td>2.4.6</td>
      <td>Results and discussions</td>
      <td>60</td>
    </tr>
    <tr>
      <td>2.4.7</td>
      <td>Comparison with other methods</td>
      <td>63</td>
    </tr>
    <tr>
      <td>2.5</td>
      <td>Conclusion</td>
      <td>64</td>
    </tr>
    <tr>
      <td><b>3</b></td>
      <td><b>Stabilized non-resonant Brillouin fiber laser</b></td>
      <td><b>67</b></td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Introduction</td>
      <td>67</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Condition for Brillouin lasing effect</td>
      <td>71</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Description of the laser architecture</td>
      <td>74</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Experimental set-up</td>
      <td>75</td>
    </tr>
    <tr>
      <td>3.4.1</td>
      <td>Characterization of Brillouin laser</td>
      <td>76</td>
    </tr>
    <tr>
      <td>3.4.2</td>
      <td>Experimental observation of mode-hopping</td>
      <td>77</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Stabilized non-resonant Brillouin fiber laser</td>
      <td>78</td>
    </tr>
    <tr>
      <td>3.6</td>
      <td>Frequency noise of non-resonant Brillouin laser</td>
      <td>81</td>
    </tr>
    <tr>
      <td>3.6.1</td>
      <td>Analytical frequency noise of non-resonant BFL</td>
      <td>82</td>
    </tr>
    <tr>
      <td>3.6.2</td>
      <td>Non-resonant BLF frequency noise using a free running pump laser</td>
      <td>84</td>
    </tr>
    <tr>
      <td>3.6.3</td>
      <td>Non-resonant BFL frequency noise using stabilized pump laser</td>
      <td>85</td>
    </tr>
    <tr>
      <td>3.6.4</td>
      <td>Frequency noise performances of two cascaded non-resonant BFL</td>
      <td>86</td>
    </tr>
    <tr>
      <td>3.7</td>
      <td>Conclusion</td>
      <td>87</td>
    </tr>
    <tr>
      <td><b>4</b></td>
      <td><b>Multi-Stokes Brillouin fiber laser</b></td>
      <td><b>89</b></td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Introduction</td>
      <td>89</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Resonant Brillouin laser</td>
      <td>90</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Brillouin laser architecture</td>
      <td>92</td>
    </tr>
    <tr>
      <td>4.3.1</td>
      <td>Fiber Ring Cavity</td>
      <td>92</td>
    </tr>
    <tr>
      <td>4.3.2</td>
      <td>Experimental setup</td>
      <td>93</td>
    </tr>
    <tr>
      <td>4.3.3</td>
      <td>Cavity servoing</td>
      <td>94</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Theoretical model</td>
      <td>94</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Power characterization of multi-Stokes Brillouin fiber laser</td>
      <td>97</td>
    </tr>
    <tr>
      <td>4.6</td>
      <td>Relative intensity noise of resonant Brillouin laser</td>
      <td>100</td>
    </tr>
    <tr>
      <td>4.6.1</td>
      <td>RIN simulations</td>
      <td>100</td>
    </tr>
    <tr>
      <td>4.6.2</td>
      <td>RIN versus the cascade process</td>
      <td>101</td>
    </tr>
    <tr>
      <td>4.6.3</td>
      <td>RIN of individual Stokes lines</td>
      <td>103</td>
    </tr>
    <tr>
      <td>4.6.3.1</td>
      <td>Single mode regime</td>
      <td>103</td>
    </tr>
    <tr>
      <td>4.6.3.2</td>
      <td>Cascaded regime</td>
      <td>105</td>
    </tr>
    <tr>
      <td>4.6.4</td>
      <td>Impact of photon lifetime</td>
      <td>107</td>
    </tr>
    <tr>
      <td>4.6.5</td>
      <td>Influence of detuning</td>
      <td>108</td>
    </tr>
    <tr>
      <td>4.6.6</td>
      <td>Conclusion</td>
      <td>111</td>
    </tr>
    <tr>
      <td>4.7</td>
      <td>Frequency noise of resonant Brillouin laser</td>
      <td>112</td>
    </tr>
    <tr>
      <td>4.7.1</td>
      <td>Short Brillouin cavity</td>
      <td>114</td>
    </tr>
    <tr>
      <td>4.7.2</td>
      <td>Frequency noise versus the cascaded process</td>
      <td>116</td>
    </tr>
    <tr>
      <td>4.7.2.1</td>
      <td>Role of anti-Stokes wave</td>
      <td>116</td>
    </tr>
    <tr>
      <td>4.7.2.2</td>
      <td>Experimental results</td>
      <td>117</td>
    </tr>
    <tr>
      <td>4.7.3</td>
      <td>Frequency noise of individual Stokes lines</td>
      <td>120</td>
    </tr>
  </tbody>
</table>
<table><tr><td>4.7.3.1</td><td>Analysis of low-Q cavity</td><td>120</td></tr><tr><td>4.7.3.2</td><td>Analysis of S1 and S2 lines</td><td>120</td></tr><tr><td>4.7.3.3</td><td>Transfer function</td><td>121</td></tr><tr><td>4.7.3.4</td><td>Frequency noise comparison of S1, S2</td><td>123</td></tr><tr><td>4.7.3.5</td><td>Conclusion</td><td>124</td></tr><tr><td>4.7.4</td><td>Impact of photon lifetime</td><td>124</td></tr><tr><td>4.8</td><td>Conclusion</td><td>126</td></tr><tr><td>A</td><td>Fresnel representation of laser noise</td><td>133</td></tr><tr><td>B</td><td>g<sub>th</sub> - factor</td><td>135</td></tr><tr><td>C</td><td>Gain distribution</td><td>137</td></tr><tr><td>D</td><td>Brillouin shift and gain detuning in multi-Stokes Brillouin laser</td><td>139</td></tr><tr><td>E</td><td>List of Publications</td><td>143</td></tr><tr><td></td><td>Bibliography</td><td>145</td></tr></table>
List of Figures

<table>
  <tbody>
    <tr>
      <td>1.1</td>
      <td>RIN measurement setup. The photocurrent from the photodetector is measured and amplified. Its DC component is blocked before its spectral analysis by the ESA. PD: photodetector, TIA: Transimpedance amplifier, ESA: Electrical spectrum analyzer.</td>
      <td>10</td>
    </tr>
    <tr>
      <td>1.2</td>
      <td>RIN of a DFB fiber laser (Koheras) emitting at a wavelength of 1550 nm</td>
      <td>12</td>
    </tr>
    <tr>
      <td>1.3</td>
      <td>Illustration of corresponding frequency noise and optical spectrum. (a) Frequency noise S<sub>ν</sub>(f) = h<sub>0</sub> = 10<sup>4</sup> Hz<sup>2</sup>/Hz. (b) Optical spectral of the laser corresponds to the white frequency noise and the intrinsic linewidth Δν<sub>L</sub> = 31.4 kHz</td>
      <td>16</td>
    </tr>
    <tr>
      <td>1.4</td>
      <td>Frequency noise and linewidth estimation using β-line method</td>
      <td>18</td>
    </tr>
    <tr>
      <td>1.5</td>
      <td>Self-heterodyne frequency noise measurement setup. OC: Optical coupler, PC: Polarization controller, AOM: Acousto-optical modulator, PD: photodetector, TIA: Transimpedance amplifier, SSA: Signal source analyzer.</td>
      <td>20</td>
    </tr>
    <tr>
      <td>1.6</td>
      <td>(a) Transfer function of an unbalanced interferometer (τ<sub>d</sub> = 200 m) (b) Cutoff frequency and gain conversion of the test bench, depending on the inserted delay length.</td>
      <td>22</td>
    </tr>
    <tr>
      <td>1.7</td>
      <td>Experimental setup for noise floor measurement. OC: Optical coupler, PC: Polarization controller, AOM: Acousto-optical modulator, PD: photodetector, TIA: Transimpedance amplifier, SSA: Signal source analyzer.</td>
      <td>22</td>
    </tr>
    <tr>
      <td>1.8</td>
      <td>(a) Phase noises of the laser and noise floors of the bench for different fiber lengths. (b) Frequency noise measurement of the Tunics OM laser and its linewidth estimation using β-line approximation.</td>
      <td>24</td>
    </tr>
    <tr>
      <td>1.9</td>
      <td>General scheme for frequency stabilization of a laser.</td>
      <td>25</td>
    </tr>
    <tr>
      <td>1.10</td>
      <td>(a) General scheme of top-of-fringe frequency stabilization. (b) Oscilloscope traces for transmission laser signal, modulation and top of fringe error signals. PD: photodetector, PID: Proportional-integral-derivative controller, VC: Variable coupler, FN: Frequency noise measurement bench.</td>
      <td>28</td>
    </tr>
    <tr>
      <td>1.11</td>
      <td>(a) General scheme of PDH frequency stabilization. (b) Experimental Oscilloscope traces for PDH error signal. PM: Phase modulator, PD: photodetector, PID: Proportional-integral-derivative controller, VC: Variable coupler, FN: Frequency noise measurement bench.</td>
      <td>29</td>
    </tr>
    <tr>
      <td>1.12</td>
      <td>(a) Frequency noise measurement of the stabilized Tunics laser on fiber ring cavity with top of fringe and PDH lock.</td>
      <td>30</td>
    </tr>
    <tr>
      <td>2.1</td>
      <td>Schematic representation of Brillouin scattering process and schematics diagram of (a) the propagation direction and (b) Momentum conservation of the incident, scattered (Stokes in orange) and acoustic wave. Red dashed lines represent anti-Stokes process.</td>
      <td>35</td>
    </tr>
  </tbody>
</table>
2.17 Fast scanning profiles recorded from the Brillouin cavity at different pumping power ($P_{in}$) with a scanning speed of 2.8 MHz/µs: The intrinsic photon lifetime ($\tau_0$), transmission and Brillouin gain coefficients are deduced from the analytical fits (red) of experimental curve (black). (a) cold cavity, (b) under coupling, (c) near to critical coupling, (d) over coupling, (f) near to transparency, (g) amplification. 60

2.18 Resonant transmission obtained from the fitting of ringing profiles in the Brillouin cavity. This figure is inspired from article [2]. 62

2.19 Brillouin gain coefficient extracted from the CRDM signal for various input pump powers. The mean value is equal to $g_B = 1.94 \times 10^{-11} \pm 1.5 \times 10^{-12} m/W$. 62

3.1 Distribution of cavity modes under the Brillouin gain curve. 72

3.2 Schematic representation of a BFL cavity with non-resonant pumping showing the different losses coefficients detailed in table (3.1). 75

3.3 Experimental setup for non-resonant fiber ring cavity Brillouin laser characterization 76

3.4 (a) Optical spectrum and (b) Power characterization of the 20 m non-resonant BFL cavity 77

3.5 Mode-hopping of non-resonant Brillouin laser cavity, observed through a Fabry-Perot analyzer in different time intervals (black signal). Red signal shows that we can observe two resonant modes at the same time under the gain bandwidth. 78

3.6 Experimental setup of BFL cavity with active PDH locking on cavity a resonances. EDFA: Erbium-doped fiber amplifier, VOA: Variable optical attenuator, OC: Optical coupler, PD: Photodetector, PM: Phase modulator, OI: Optical isolator, OF: Optical filter. 79

3.7 (a) Beating between pump and Stokes wave with and without mode-hopping stabilization scheme. (b) Optical spectrum of stabilized non-resonant BFL cavity. 80

3.8 Analytical phase noise transfer function for the 20 m fiber length cavity with $R = 0.78$ and $\beta_A = 17.27$. 82

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. 84

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. 86

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. 91

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. 93
<table>
  <tbody>
    <tr>
      <td>4.3</td>
      <td>Steady-state laser powers for the Stokes components, obtained from the simulation.</td>
      <td>96</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>(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.</td>
      <td>98</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Power characterization of MBL<sub>HQ</sub> cavity.</td>
      <td>98</td>
    </tr>
    <tr>
      <td>4.6</td>
      <td>Left axis: Output Stokes power (P<sub>s</sub>) normalized to the circulating power (P<sub>c</sub>) versus input pump power (P<sub>in</sub>) normalized to the Stokes 1 lasing threshold (P<sub>th</sub>). 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<sub>th</sub> = 26.5 mW.</td>
      <td>102</td>
    </tr>
    <tr>
      <td>4.7</td>
      <td>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.</td>
      <td>104</td>
    </tr>
    <tr>
      <td>4.8</td>
      <td>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.</td>
      <td>105</td>
    </tr>
    <tr>
      <td>4.9</td>
      <td>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 refer to high (τ<sub>0</sub> = 1.4 µs, τ<sub>e</sub> = 3.7 µs) and low-Q factors (τ<sub>0</sub> = 1.4 µs, τ<sub>e</sub> = 0.5 µs) respectively.</td>
      <td>106</td>
    </tr>
    <tr>
      <td>4.10</td>
      <td>Schematic presentation of frequency fluctuations converted into intensity fluctuations cavity Q-factors.</td>
      <td>107</td>
    </tr>
    <tr>
      <td>4.11</td>
      <td>Schematics presentation of gain detuning of Brillouin cavity.</td>
      <td>109</td>
    </tr>
    <tr>
      <td>4.12</td>
      <td>Evolution of the Stokes 1 RIN as a function of the gain detuning for a pump power of a) 5.5 P<sub>th</sub> and b) 9 P<sub>th</sub> in the low-Q cavity configuration.</td>
      <td>110</td>
    </tr>
    <tr>
      <td>4.13</td>
      <td>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.</td>
      <td>111</td>
    </tr>
    <tr>
      <td>4.14</td>
      <td>Experimental setup of the BFL cavity with PDH locking.</td>
      <td>114</td>
    </tr>
    <tr>
      <td>4.15</td>
      <td>BFL detuning measurement: The measured Brillouin frequency shift (squares) and S1 threshold (dot) are plotted as a function of pump wavelength.</td>
      <td>115</td>
    </tr>
    <tr>
      <td>4.16</td>
      <td>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).</td>
      <td>117</td>
    </tr>
    <tr>
      <td>4.17</td>
      <td>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).</td>
      <td>118</td>
    </tr>
    <tr>
      <td>4.18</td>
      <td>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.</td>
      <td>118</td>
    </tr>
    <tr>
      <td>4.19</td>
      <td>(a) S1 frequency noise (b) S2 frequency noise for different regimes.</td>
      <td>121</td>
    </tr>
    <tr>
      <td>4.20</td>
      <td>The pump frequency noise to Stokes frequency noise (a) zero detuning (b) with pump and SBS gain detuning [4].</td>
      <td>122</td>
    </tr>
    <tr>
      <td>4.21</td>
      <td>Comparison of frequency noise of S1 and S2 at 8.3 P<sub>th</sub></td>
      <td>123</td>
    </tr>
  </tbody>
</table>
List of Tables

<table><tbody><tr><td>2.1</td><td>Parameters obtained from the fitting of ringing profiles at different pump powers</td><td>61</td></tr><tr><td>2.2</td><td>Comparison of g<sub>B</sub> values obtained for single mode silica fibers [5] in various works. SH stands for self-heterodyne.</td><td>63</td></tr><tr><td>2.3</td><td>Material values for pure and 3% GeO<sub>2</sub> doped silica fibers.</td><td>63</td></tr><tr><td>2.4</td><td>Parameters of a 3% GeO<sub>2</sub> doped silica fibers</td><td>64</td></tr><tr><td>3.1</td><td>Losses coefficients of each components in the cavity</td><td>75</td></tr><tr><td>3.2</td><td>Parameters of the non-resonant BFL cavity</td><td>76</td></tr><tr><td>4.1</td><td>Parameters of resonant BFL 20m cavity</td><td>93</td></tr><tr><td>4.2</td><td>Threshold of resonant BFL cavity</td><td>99</td></tr><tr><td>4.3</td><td>Comparison of S1 linewidth in different cavities</td><td>125</td></tr><tr><td>D.1</td><td>Gain detuning in cascaded Brillouin laser</td><td>142</td></tr></tbody></table>
# Introduction

In 1922, Léon Brillouin theoretically predicted the scattering due to the interaction of the photons and acoustic waves (propagating density fluctuations) [6]. The experimental verification was difficult at that time due to the lack of coherent light source to efficiently generate an ultrasonic pressure wave. In 1930, Gross [7] experimentally proved the existence of photon and acoustics wave interaction. Gross was originally looking for the Raman scattering in organic liquids, but his measurements did not agree with Raman scattering. However it perfectly matched the theoretical predictions made by Brillouin. Note that Mandelstam predicted a similar scattering effect in 1926 [8]. Based on the work of Leon Brillouin, the inelastic scattering of the light from acoustic waves is referred as Brillouin scattering in literature.

The discovery of the laser (light amplification by stimulated emission of radiation) in the 1960s, revolutionized the study of fundamental physics by giving birth to new research like laser physics, nonlinear optics, etc. The lasers provide a coherent and powerful light source, that leads to changes in the material properties. Nowadays, lasers play a vital role in telecommunication, medicine, defense, industry, environment, lightning ... In recent decades, different types of lasers have been developed in terms of optical power, coherence and wavelengths.

Soon after the first realization of a laser [9], Chaio and Townes observed, in 1964, stimulated Brillouin scattering (SBS) in quartz and sapphire [10]. After the invention of optical fibers, stimulated Brillouin scattering was first studied in 1972 by Ippen and Stolen [11]. Then SBS gained attention in telecommunication systems because it limits the performances by reflecting most of the power through the counter-propagating Stokes wave. It results in large depletion of pump power. On the contrary SBS can be used on purposes as optical amplifiers [12], lasers [13], filters [14, 15], and sensors [16, 17]. Informations from the SBS may give the mechanical properties of materials, which open
the field of Brillouin based bio-imaging [18, 19, 20]. In the past century, a tremendous amount of research was done concerning Brillouin scattering, especially in the last decades during which, SBS was observed in chip-scale structures [21, 22], and different materials like silica [11], chalcogenide [23], tellurium [24].

In 1976, Hill [13] realized a fiber laser based on Brillouin effects, which had the unique ability to reduce noise compared to that of the pump [25]. Later, different architectures of Brillouin lasers [26, 25, 27, 28, 29, 30, 31, 32] are studied. This low-noise laser is of interest to the scientific community for applications in microwave and optical signal processing, distributed sensors, spectroscopy, gyroscope, metrology. The noise properties of this laser are not well understood. In this thesis, we explain two Brillouin laser architectures: resonant and non-resonant Brillouin laser and their noise properties.

## Objectives of the thesis

This thesis is devoted to the understanding of ring Brillouin fiber lasers, operating with multiple Stokes orders. Most of the work carried out during this thesis is part of the continuity of the works of Kenny Tow [33] and Schadrac Fresnel [34]. My work was accomplished in the framework of the SOLBO project (SOurces Laser Brillouin cOhérent à faible bruit) of the FUI call (Fonds Unique Interministériel AAP20) with the help of BPI France, the "Images et réseaux" cluster and in collaboration with IDIL Fibre Optiques (project leader), Lumibird, THALES, ISCR (Institut des Sciences Chimiques de Rennes), Photonics Bretagne.

The main objective was to design a low cost Brillouin fiber laser, with an integrated linewidth below kHz, by IDIL and also for the Foton Institute, to explore the physics of multi-Stokes Brillouin fiber lasers and to try to know the fundamental limitations of frequency noise reduction for these sources, whether for single or multiple Stokes order cavities. In this thesis, I propose

* a new Brillouin gain measurement technique, particularly adapted to short cavities and not requiring high optical pump powers (a few tens of milliWatts),

* a comparison of laser operation between resonant and non-resonant pumping,

* a modeling of our experimental results by an analytical approach, allowing us to explain the physical mechanisms of these lasers,
• a discussion of the fundamental limitations of these devices and their advantages.

Main results

Firstly, we will report a technique based upon the cavity ring-down method that enables the characterization of the Brillouin gain coefficient directly from probing the laser cavity. Material gain, optical cavity parameters, and lasing properties can be extracted from measurements within a single experiment. This method could be used to characterize an integrated microresonator Brillouin cavity. Secondly, we will study two different Brillouin designs, which are using resonant or non-resonant pumping. In non-resonant Brillouin laser, the pump is doing a single pass inside the cavity, and the generated Stokes wave are resonant inside the cavity. In order to increase the long-term stability of such scheme, we will propose a frequency stabilized non-resonant Brillouin fiber laser design, that enables us to compare our analytical and experimental results about frequency noise in a non-resonant Brillouin laser. The intrinsic linewidth of our non-resonant Brillouin laser is 1.3 Hz. Furthermore, we demonstrate experimentally that the intrinsic linewidth can be reduced down to 47 mHz by cascading two non-resonant Brillouin lasers, showing the ease and ability of such systems to produce low-noise light. Unfortunately, we did not succeed to explore the fundamental limitations of noise reduction due to the noise floor of our bench.

A major work of this thesis is the investigation of noise properties of a resonant
Brillouin laser. For resonant pumping, the pump and Stokes waves are resonant inside
the cavity, which could produce multi-Stokes orders. We experimentally analyze, in this
cascaded regime, steady-state optical powers and noise properties of the different signals,
including frequency noise and relative intensity noise. Our experimental results reveal
that the cascading process modifies the dynamics of Brillouin laser when compared to
those of single-mode Brillouin laser with a single first order Stokes component.

We analytically and experimentally investigate the intensity noise of each Stokes wave
and study the noise dynamics of the cascaded Brillouin scattering process. In particular
we examine the impact of fiber ring quality factor on the laser RIN features such as
amplitude-noise reduction and relaxation frequency. We also investigate the influence of
the Brillouin gain detuning on the RIN reduction. This detailed study is important as
many characteristics of frequency noise are related to RIN properties through a coupling
between intensity and phase.

The study of frequency noise dynamics of a multi-Stokes Brillouin fiber laser was
done on a large mode-volume optical resonator with 2 m optical cavity, because this kind
of cavity is less sensitive to acoustical and thermal fluctuations than micro-cavity. It
will enable to reach lower flicker noise. With this cavity, we have reached an intrinsic
linewidth of 12.8 Hz and less than 2 kHz integrated linewidth for 10 ms observation time.

Furthermore, we will experimentally demonstrate for the first time that the frequency
noise is increased in the presence of anti-Stokes Brillouin scattering in the cascading
process. We will also show that the gain detuning could enhance the linewidth by an
amplitude-phase coupling. All these experimental results are in good agreement with our
analytical calculations obtained through our nonlinear system for multi-Stokes Brillouin
laser. This good match is mainly due to three facts:

1. we are able to get directly very precise values of cavity parameters and of the Brillouin gain coefficient using the CRDM technique;

2. we are able to reach long term stability (tens of hours);

3. we are able to finely control the detuning between the laser Stoke resonance and the frequency of Brillouin gain maximum.

All these very fine observations allow us to design and engineer Brillouin fiber lasers.
We show some fundamental limitations of these systems in terms of noise reduction such
as

• the role of pump noise, possibly in its interrelation with cavity finesse,

• anti-Stokes orders,

• the effect of detuning is inherent in higher Stokes orders.

Nevertheless, such systems are still very pervasive as evidenced by the interest they arouse
in the scientific community. This Ph.D. thesis contributes to a better understanding of
multi-Stokes Brillouin lasers.
# Thesis organization

This thesis is organized into four chapters followed by a conclusion and some remarks.
In chapter 1, we give a general introduction to the physics of laser noise necessary to understand the work of this manuscript. It also details the measurement technique for frequency noise and relative intensity noise (RIN) of laser. Moreover, we introduce fundamental ingredients of the laser stabilization methods, which have been used for the realization of Brillouin laser.

Chapter 2 begins by describing physical mechanisms behind the Brillouin scattering in optical fibers. We will briefly describe spontaneous and stimulated Brillouin scattering and will explain a new Brillouin gain characterization based on the cavity ring-down method (CRDM). We will also compare the results with conventional Brillouin gain characterization. This cavity ring-down method enables us to identify the different coupling regimes and estimate the Q-factor of the resonator. These measurements are fundamental to characterize Brillouin laser cavity and ease to design Brillouin cavity.

In chapter 3, we will describe a stabilized non-resonant Brillouin laser. The chapter begins with a historical overview of Brillouin lasers. This overview explores different laser designs and current state of the art of Brillouin lasers in fiber, integrated chip, and microresonator. Moreover, we address the issue of mode-hopping in non-resonant Brillouin laser and propose a frequency stabilization method based on Pound-Drever-Hall (PDH) locking. Also, we investigate the power and frequency noise characterization of non-resonant Brillouin laser. We show experimental frequency noise agreement to analytical results and corresponding laser linewidth are in the range of Hz level. We demonstrate that the cascading of two Brillouin lasers can reach the mHz linewidth.

In chapter 4, we will explore the physics of resonant Brillouin laser. We will discuss the power and noise dynamics of these lasers with the help of our numerical and experimental results. RIN of resonant Brillouin lasers will be studied in terms of pumping power, photon lifetime, and gain detuning in the so-called single-mode and cascading regimes of the laser. We will show that our numerical simulations and experimental results are in good agreement. We will discuss some fundamental limitations of noise reduction linked to pump noise, Q-factor, anti-Stokes orders, gain detuning and amplitude phase-coupling.
<figure><img src="image_3.png" /></figure>
# Chapter 1

## Noise in lasers

### 1.1 Introduction

Narrow linewidth lasers are of great importance for several areas, such as coherent optical telecommunications [35, 36], sensors [37], precision spectroscopy [38], optical atomic clocks [39, 40], frequency metrology [41, 42], LiDAR [43] and microwave photonics [44] to only name a few. My work is mainly focused on the development of low-noise Brillouin laser; the study of these lasers demonstrates that it is possible to produce a laser output with low-noises when compared to those of the pump laser. But before explaining this, it is necessary to introduce the theoretical and experimental basics of laser noises, which are available in many books [33, 34, 45, 46, 47, 48, 49, 50]. The object of this chapter is to introduce laser noises descriptions. The origin of the laser noise is defined and introduced, in section (1.2), by considering the amplitude and phase noises. In section (1.3), theory and experimental measurement methods are explained for laser relative intensity noise (RIN) while section (1.4) is dedicated to laser phase noise. Moreover, theoretical and experimental basics of the linewidth narrowing method, which is based upon laser frequency-stabilization on fiber ring cavity are introduced in section (1.5).

### 1.2 Origin of noise in lasers

In a laser, spontaneous emission plays an essential role in the generation of initial photons that leads to amplification by stimulated emission, and that will give the laser emission. The spontaneous emission is a random process, which emits photons with
random direction, phase and polarization. These random fluctuations are called intrinsic or fundamental "noise" of the laser. These fluctuations may have a negative impact on numerous laser applications, more particularly in the field of metrology and optical sensors.

In presence of noise, the scalar electric field associated with the light wave, delivered by a single mode laser at the optical frequency of $\nu_0$, is written:

$$E(t) = [E_0 + \delta E(t)]e^{i[2\pi\nu_0 t + \phi(t)]} \quad (1.1)$$

Where $\delta E(t)$ and $\phi(t)$ are temporal fluctuations of amplitude and phase. The representation is linked to the Fresnel vector in the complex plane (It is explained in appendix (A) [50]). The origin of these fluctuations can be classified into two categories: the first one is at the quantum level; this fundamental noise comes from the spontaneous emission process in lasers. The electromagnetic thermal radiation will also contribute to fluctuations. The second one is called technical noise; it may include several contributions as: acoustical and mechanical vibrations, temperature fluctuations or even the pump noise. These various random contributions lead to amplitude and phase noises that will be studied through their power spectral density (PSD).

PSD of a given random variable, $\delta b(t)$ (to be more general), will be the Fourier transform of its autocorrelation function $R_{\delta b}$. $b$ can be either intensity or phase in the present study. The PSD can be written as $S_{\delta b}$:

$$S_{\delta b}(f) = \int_{-\infty}^{+\infty} R_{\delta b}(\tau)e^{-i2\pi f\tau}d\tau \quad (1.2)$$

with $R_{\delta b}$ being the autocorrelation function of $\delta b(t)$ defined by

$$R_{\delta b}(\tau) = \langle \delta b(t) \delta b(t - \tau) \rangle = \int_{-\infty}^{+\infty} \delta b(t) \delta b(t - \tau) dt \quad (1.3)$$

Note that in Eq. (1.2) the frequencies of interest are mainly in the electrical domain [DC to a few GHz]. For that reason $f$ may be used instead of $\nu$ to indicate low frequency components. It's why the measurements are carried out in the electrical field rather than in the optical field by an electrical spectrum analyzer (ESA). In the next sections, we will define amplitude and phase noises and explain the associated measurement methods to characterize them.
1.3 Relative intensity noise (RIN)

One of the objectives of this thesis is to study the intensity noise of a Brillouin laser experimentally and theoretically. In particular, we experimentally analyze the intensity noise of each Stokes wave generated through the Brillouin process. We examine the impact of the cavity parameters on the laser RIN features. In this section, we introduce the RIN principles and measurement method, some of the notations and explanations are taken from the following references [33, 34].

1.3.1 Definition of relative intensity noise

The emitted laser field (Eq.1.1) has temporal amplitude fluctuations; the related optical power as well:

$$
P(t) = P_0 + \delta P(t) \tag{1.4}
$$

where $P_0$ is the average power and $\delta P(t)$ is the temporal fluctuation of optical power. It describes the fluctuations of the optical power output of the laser around its average power of $P_0$. The intensity noise is obtained by dividing the optical power by the effective area, which is the illuminated surface. Then the laser relative intensity noise can be defined by taking a Fourier transform of $\delta P(t)$

$$
RIN(f) = \frac{\langle |\delta P(f)|^2 \rangle}{P_0^2} \qquad (1.5)
$$

$\langle \rangle$ stands for statistical expectation. Thus the RIN is defined by the ratio of the power spectral density $S_{\delta P}$<fnref n="1" /> over the average optical power. $S_{\delta P}$ is the Fourier transform of the auto-correlation of $\delta P(t)$ and is related to the second moment of the noise forces. The definition can be taken in the electrical domain:

$$
RIN(f) = \frac{S_{RIN}}{P_{ele}} \qquad (1.6)
$$

Where $S_{RIN}$ is the electrical power spectral density. $P_{ele}$ is average electrical power.
The relation between optical power and electric power $P_{ele} = R I_0^2 = R \eta^2 P_0^2$; R is the
load resistance of the photodetector. $I_0$ is the average photocurrent produced by the
photodetector. $\eta$ is the quantum efficiency of the photodetector. In the following, the
RIN measurement method used in this manuscript is described.

<footnote n="1">¹$S_{\delta P}$ can be obtained from Eq (1.2) by considering the $R_{\delta P}(\tau) = \langle \delta P(t) \delta P(t - \tau) \rangle$</footnote>
### 1.3.2 RIN measurement method

At Institut FOTON, we study the intensity noise by a direct detection of PSD using an electrical spectrum analyzer [33, 34]. The experimental setup used to characterize the RIN of a laser during this thesis is shown in figure (1.1).

<figure><img src="image_4.png" /><figcaption>Figure 1.1: RIN measurement setup. The photocurrent from the photodetector is measured and amplified. Its DC component is blocked before its spectral analysis by the ESA. PD: photodetector, TIA: Transimpedance amplifier, ESA: Electrical spectrum analyzer.</figcaption></figure>

The RIN detection system consists of a photodetector with a bandwidth from DC to 1 GHz, a trans-impedance amplifier (TIA) with a variable bandwidth depending on the gain, but not exceeding 200 MHz, and a "DC-Block" module with very low cut-off frequency (1 Hz), to remove the DC component of the electrical signal in order to avoid damaging the ESA. PSD is analyzed with the help of the ESA. The PSD collected on the ESA contains not only the PSD of the RIN but also that of thermal noise ($S_T$) and shot noise ($S_{sh}$). The PSD detected at the electric spectrum analyzer is then the sum of these three noise sources. As they are independent stochastic processes, the total PSD can be written as:

$$S_{Total} = S_{T} + S_{sh} + S_{RIN} \qquad (1.7)$$

It is then necessary to measure the thermal and shot noise contributions in order to obtain the laser RIN. It will be explained in the next section.

### 1.3.3 Determination of thermal noise

Thermal noise $S_T$ is independent of the optical signal received on the detector but depends on the devices in the measurement system (ESA, photodetector, amplifier). It is simply measured without an optical signal on photodetector. It consists of two main contributions: thermal noise (or Johnson noise or Johnson-Nyquist noise) and flicker, called also excess or $1/f$ noise. In the case of a photodetection system, even in the absence of
the incident signal, a current is generated by the movement of the charge carriers in the material due to thermal agitation. These current fluctuations induce thermal noise. The PSD of fluctuations due to thermal noise can be expressed as

$$S_T = 4k_B T B \qquad (1.8)$$

where $k_B = 1.3807 \cdot 10^{-23} J/K$ is the Boltzmann constant, T is the temperature given in Kelvin and B is the detection bandwidth. The PSD of the flicker noise is linked to electronic components (photodetector, amplifiers, ESA) that are used to measure. This noise is the consequence of several random processes [51]: fluctuations in the number of carriers [52], fluctuations in the mobility of carriers [53]. It is due to the presence of impurities and structural defects during the manufacture of electronic components. So this noise depends on the quality of the manufacturing and the technology employed. In the case of semiconductors, this noise can become annoying for frequencies below a few kHz. Thermal noise is the minimum signal level measurable by the detection bench and then have to be reduced by a judicious choice of devices.

### 1.3.4 Determination of shot noise

Shot noise ($S_{sh}$) is related to the randomness of the photons and electrons during the creation of photocurrent in the photodetector. The photocurrent is a function of the incident optical power. The PSD of shot noise is related to the photocurrent. If $H(f)$ is the transfer function of the detection chain, then the PSD of the shot noise is written as [54]:

$$S_{sh} = |H(f)|^2 2qRI_0 \qquad (1.9)$$

where R is the load resistance of the spectrum analyzer. q is the electronic charge equal to $1.602 \times 10^{-19}$ coulomb, and $I_0$ is the average photocurrent.

To evaluate the shot noise, it is necessary to use a reference source having a negligible intensity noise with respect to that of the shot noise [55]. White light sources are good candidates for this measurement [33, 34]. To ensure that the detection is at the level of the shot noise, we check the linearity of the measurements with the photocurrent. If the relative intensity noise is not negligible, we will have a non-linear behavior. Quantitatively, the shot noise PSD normalized by the photocurrent is given by the relation:

$$S_{shnorm} = |H(f)|^2 2qR \qquad (1.10)$$
Then the transfer function of the detection system can be written as:

$$|H(f)|^2 = \frac{S_{shnorm}}{2qR} \qquad (1.11)$$

From these thermal and shot noise PSD determination, we can calculate the RIN, as it will be explained in the next section.

### 1.3.5 Determination of relative intensity noise

The spectral density of the laser intensity noise is given by the following relation:

$$S_{RIN} = |H(f)|^2 RI^2 10^{RIN/10} \qquad (1.12)$$

RIN is expressed in dBc/Hz and by abuse of use dB/Hz is used. The RIN of a laser can be obtained by insertion of Eqs. (1.12), (1.11), (1.10) in Eq (1.7):

$$RIN = 10 \log_{10} \left( \frac{2q}{I} \left[ \frac{S_{Total} - S_{T}}{S_{shnorm} I} - 1 \right] \right) \qquad (1.13)$$

Figure (1.2) shows an example of RIN spectrum of a DFB fiber laser (Koheras<sup>TM</sup>) emitting at a wavelength of 1.55 µm that will be used as a pump laser in the following chapters of this thesis. The peak around 154 kHz corresponds to the relaxation frequency of the laser. Beyond this frequency, the PSD decreases very quickly with a slope of 20 dB per decade. RIN is usually low in laser due to the clamping of the gain that reduces

<figure><img src="image_5.png" /><figcaption>Figure 1.2: RIN of a DFB fiber laser (Koheras) emitting at a wavelength of 1550 nm</figcaption></figure>

the input pump noise. This feedback mechanism is not present on the phase of the laser that continuously drifts and implies a larger noise contribution than in amplitude. The phase noise is explained in next sections.
1.4 Phase noise

In this section, we review the basic concepts of phase and frequency noises of a laser. We also describe the relations between the phase noise, frequency noise, and spectral laser linewidth. Moreover, we explain the measurement method of laser frequency noise based on the self heterodyne technique.

As we have seen in section (1.2), if we consider the laser emission as a perfect signal $E(t) = E_0 e^{i[2\pi\nu_0 t]}$, then the power is concentrated at a carrier frequency $\nu_0$, the optical spectrum will be a Dirac function. But as a matter of fact, a sinusoidal signal has infinite energy and a more precise description will consider a broader emission with finite energy. Then, laser signal will fluctuate randomly in phase so the total power of the signal is symmetrically distributed on each side of the frequency $\nu_0$. The full width at half maximum (FWHM) of the energy distribution is called the laser linewidth. According to the theory of Schawlow and Townes [56], the linewidth of a laser is due to phase noise caused by spontaneous emission of photons in a laser cavity. This fundamental limit of a laser linewidth can be described by a Lorentzian spectral lineshape:

$$ \Delta\nu_L = \frac{\pi h\nu}{(2\delta_{1/2})^2 P_0} \qquad (1.14) $$

Where $h\nu$ is the photon energy, $2\delta_{1/2}$ is the laser cavity resonance width and the $P_0$ is the laser output power. Generally, this value is very difficult to measure from the optical spectrum because other technical noise may increase the laser linewidth such as acoustical and mechanical vibrations, mode beatings between different longitudinal or transverse modes and noise transfer from the pump sources to the laser. In semiconductor laser this linewidth will be higher because of the coupling between intensity and phase, which is caused by the refractive index dependence on the carrier density. The study of the phase or frequency noises will give information about the Lorentzian linewidth of the laser and other contributions like Gaussian distribution due to environmental noises. Laser phase noise is characterized in the electrical domain with a single side-band power spectral density $S_\phi(f)$. In the following subsections, we will firstly introduce the relation between phase and frequency noise. Then we will explain the linewidth determination from the frequency noise underlying different noises like white and flicker noise in section (1.4.2). Section (1.4.3) details the experimental method to characterize the frequency
noise using the self-heterodyne method. The inherent noise floor of the experimental bench is analyzed.

1.4.1 Frequency noise in laser

The spectral purity of the laser can be expressed as a function of frequency. The instan-
taneous frequency of the laser is:

$$
\nu(t) = \nu_0 + \delta\nu(t) \tag{1.15}
$$

Where $\nu_0$ is the central frequency. $\delta\nu(t)$ is related to the fluctuation of the frequency and phase by $\delta\nu(t) = \frac{1}{2\pi} \frac{d\phi(t)}{dt}$.

$$
\nu(t) = \frac{1}{2\pi} \frac{d}{dt} [2\pi\nu_0 t + \phi(t)] = \nu_0 + \frac{1}{2\pi} \frac{d\phi(t)}{dt} \quad (1.16)
$$

PSD of the frequency fluctuations $\delta\nu(t)$ can be obtained by Eq (1.17):

$$
S_{\nu}(f) = \int_{-\infty}^{+\infty} R_{\nu(t)}(\tau) e^{-i2\pi f\tau} d\tau \quad (1.17)
$$

with $R_\nu(\tau)$ being the autocorrelation function of $\delta\nu(t)$ defined by

$$
R_{\nu}(\tau) = \langle \delta\nu(t)\delta\nu(t-\tau) \rangle = \int_{-\infty}^{+\infty} \delta\nu(t)\delta\nu(t-\tau)dt \quad (1.18)
$$

$S_\nu(f)$ is expressed in $Hz^2/Hz$. The relation between the phase and frequency noise is then linked through a derivative in the time domain, which implies, for the PSD, a proportional factor equal to the square of the frequency:

$$
S_{\nu}(f) = f^{2} S_{\phi}(f) \tag{1.19}
$$

In general, PSD of laser frequency noise can be expressed in the form :

$$
S_{\nu}(f) = \frac{h_{-1}}{f} + h_{0} \tag{1.20}
$$

Where $h_0$ is a white noise contribution (due to spontaneous emission). $\frac{h_{-1}}{f}$ is the laser flicker noise. These two contributions are the main ones in lasers. However the -1 term may be replaced by $\frac{h_{-2}}{f^2}$, which is linked to a random walk frequency noise (a term $\frac{h_{-\alpha}}{f^\alpha}$ with $1 < \alpha < 2$, may be also added or it may replace $\frac{h_{-1}}{f}$). It is interesting to know how to relate the frequency noise obtained from the noise PSD and the laser linewidth [57, 58, 59] (that might be obtained from the optical spectrum).
### 1.4.2 Relation between the power spectral density of frequency noise and optical spectrum

In this section we present a simple approach to estimate the laser linewidth from the frequency noise, a detailed explanation can be found in [57, 58, 59]. If we consider that the laser field is written as $E(t) = E_0e^{i(2\pi\nu_0t+\phi(t))}$, neglecting the intensity noise $\delta E(t) = 0$, then the PSD of its frequency noise is linked to the auto-correlation function as follows:

$$R_E(\tau) = E_0^2 e^{i2\pi\nu_0\tau} e^{-2\int_0^{+\infty} S_\nu(f) \frac{\sin^2(\pi f \tau)}{f^2} df} \quad (1.21)$$

The laser spectrum is obtained by performing a Fourier transform:

$$S_E(\nu) = 2 \int_{-\infty}^{+\infty} e^{-i2\pi\nu\tau} R_E(\tau) d\tau \quad (1.22)$$

As we have seen before a laser frequency noise PSD may be the sum of three types of noise: white, flicker, and random walk. In this thesis, we are focused only on white and flicker noises. In the following, we will detail the contributions of the white and flicker noises separately. Finally, we will explain the general case, which includes both noise contributions in the laser spectrum. The linewidth associated with the white noise is called intrinsic linewidth.

#### White noise

Let consider that the laser emission is only under the influence of spontaneous emission (white noise), then the PSD of the frequency noise (Eq 1.20) reduces to $S_\nu(f) = h_0$. The optical spectrum Eq (1.22) is then written as:

$$S_E(\nu) = E_0^2 \frac{h_0}{(\nu - \nu_0)^2 + (\pi h_0 / 2)^2} \quad (1.23)$$

The laser spectrum has a Lorentzian shape. Its FWHM is the laser intrinsic linewidth. $\Delta\nu_L = \pi h_0$. In figure (1.3) plotted the white frequency noise and its optical spectrum. The white frequency noise is plotted for a value $h_0 = 10^4 \text{ Hz}^2/\text{Hz}$ corresponding Lorentzian shape spectrum is give a intrinsic linewidth $\Delta\nu_L = 31.4 \text{ kHz}$. In this thesis manuscript, this relation is used to determine the intrinsic linewidth of a Brillouin laser.

#### Flicker noise
<figure><img src="image_6.png" /><figcaption>Figure 1.3: Illustration of corresponding frequency noise and optical spectrum. (a) Frequency noise $S_ν(f) = h_0 = 10^4 Hz^2/Hz$. (b) Optical spectral of the laser corresponds to the white frequency noise and the intrinsic linewidth $\Delta\nu_L = 31.4 kHz$</figcaption></figure>

Frequency flicker noise is slightly more difficult to handle than in the case of white noise. Considering the flicker noise PSD is $S_ν(f) = \frac{h_{-1}}{f}$. Then Eq. (1.21) is written as

$$R_E(\tau) = E_0^2 e^{i2\pi\nu_0\tau} e^{-2\int_0^{+\infty} h_{-1}(f) \frac{\sin^2(\pi f \tau)}{f^3} df} \quad (1.24)$$

In this expression we can see that the integral ($J = \int_0^{+\infty} h_{-1}(f) \frac{\sin^2(\pi f \tau)}{f^3} df$) is not defined at $f = 0$ but it can be shown [57] that it is a finite integral with an asymptotic value $\sigma^2 = 3.56h_{-1}$, in which $\sigma^2$ is linked to the width of the integrated linewidth.

In order to simplify this problem, we may introduce the notion of observation time ($T_{obs}$) and the associated minimum frequency (which is simply linked through an inverse relationship). For example, if we measure the frequency noise of a laser at 1 kHz from the carrier then the observation time will be 1 ms (1/1 kHz). The optical spectrum of the flicker noise is a Gaussian function [57].

$$S_E(\nu) = E_0^2 \frac{\sqrt{\pi}}{\sigma} e^{-(\nu_0 - \nu)^2 / \sigma^2} \quad (1.25)$$

This Gaussian function is centered on $\nu_0$ and corresponds to the density probability function of the laser frequency. In other words, the laser frequency may have deviations from its central frequency $\nu_0$. In the case of flicker noise, the mean deviation tends to an asymptotic value for infinite time of observation, which is linked to the fact that as shown above $\sigma^2 = 3.56h_{-1}$. The linewidth associated with the terms $h_{-1}$, which implies a frequency jitter, is called integrated linewidth.
## Case of laser with flicker and white noise

In the general case, the laser frequency noise contains both white and flicker noise as explained in Eq. (1.20). Then the laser optical spectrum is a Voigt function, which is the convolution of a Lorentzian (white noise) and a Gaussian (flicker noise) functions [58]:

$$S_E(\nu) = E_0^2 \frac{\Gamma}{\sqrt{\pi}\sigma^2} \int_{+\infty}^{-\infty} \frac{e^{-t^2}}{(X^2 - t)^2 + Y^2} dt \quad (1.26)$$

$$X = \frac{\nu - \nu_0}{\sigma}, \quad Y = \frac{\Gamma}{\sigma}, \quad t = \frac{\sigma\tau}{2} + iX + Y, \quad \Gamma = 2\pi^2 h_0 \quad (1.27)$$

It follows that the linewidth of the Voigt function will depend on the observation time as it has been underlined for the Gaussian function linewidth in the previous section. The Lorentzian function corresponds to the spontaneous emission, which is a white noise ($h_0$). The central frequency of the Lorentzian contribution will deviate in time with a probability given by a Gaussian function (linked to $h_{-1}$ as mentioned above).

## β-line approximation

The lasers that we use in this thesis, has white and technical frequency noise, which can be either a flicker or random walk noise. As we have seen previously the intrinsic linewidth of the laser is easily estimated from the white frequency noise. To estimate the value of the linewidth given by Eq. (1.21) from an experimental frequency noise spectrum, we opted for a recently proposed approach in the literature [57]. It assumes that in practice, any PSD of the frequency noise is separated into two regions by drawing a line, which is called "β-line" (see figure (1.4)). This β-line is described by:

$$S_{\beta-line}(f) = \frac{8\ln(2)}{\pi^2} f \quad (1.28)$$

The intersection of the line $S_{\beta-line}$ and of the PSD is at a frequency $f_{\beta-line}$ that indicates the separation of the two regions. The method states that: frequency noise located below the line $f_{\beta-line}$ is usually a white frequency noise due to spontaneous emission. It implies a Lorentzian line-shape. White noise at higher frequencies contributes only to the wings of the optical spectrum, when considering the Voigt profile. The frequency noise located below the $f_{\beta-line}$ frequency will contribute to the Gaussian shape and it will depend on
the observation time. The linewidth of the optical spectrum can be estimated with a 10 % error by the following equation.

$$ \Delta\nu_g = \sqrt{8\ln(2)}A \qquad (1.29) $$

Where A is the area under the curve:

$$ A = \int_{1/T_{\text{obs}}}^{f_{\beta-line}} S_{\nu}(f) df \qquad (1.30) $$

Figure (1.4) shows an example of the β-line approximation. Laser frequency noise given

<figure><img src="image_7.png" /><figcaption>Figure 1.4: Frequency noise and linewidth estimation using β-line method</figcaption></figure>

by Eq (1.20) is plotted as a function of the frequency for values $h_{-1} = 1 \times 10^7(Hz^2)$ and $h_0 = 1 \times 10^4(Hz^2/Hz)$. The red dashed line represents the β-line and it intersects the frequency noise at $f_{β-line}$. The shaded region is the area under the curve, which gives the contribution to the Gaussian line shape (or the low-frequency contribution, implying a frequency jitter of the central frequency of the Lorentzian laser line). Integrated linewidth for an observation time of 10 ms ($1/T_{\text{obs}} = 100$ Hz) gives 37.1 kHz. The intrinsic linewidth is $\pi h_0 = 31.4$ kHz. This method enables to give an estimated linewidth, which could be useful in addition to a direct reading of the frequency noise. It gives a good estimation of the integrated linewidth for a given time of observation. In other words, it is an estimation of the impact of the low frequency noise (flicker ...). This method allows to evaluate the linewidth of a source which features a $1/f$ frequency noise component and a white noise component. This is the case of many lasers, externally stabilized on a cavity or not. The stabilization reduces the noise but there is always $1/f$ noise close
to the carrier. Sometimes the shape of the frequency noise spectrum is more complex
than this two slopes description and the beta-line approximation cannot be used. The
frequency noise is measured using a correlated delayed self-heterodyne method, which
will be explained in the next section.

1.4.3 Experimental characterization of frequency noise

As we have seen in the previous subsection, the phase/frequency noise enables to analyze
laser emission. In this section, we will explain the method, used in our laboratory, to
measure frequency noise. As previously mentioned, phase and frequency noise are related.
Generally, phase noise measurements are classified into three categories.

The first one is a direct heterodyne detection of two lasers [41, 60]. The signal of the laser under test is beating with the one of a reference laser, for which the phase noise is much lower than that of the laser under test. The resulting beat-note signal transfer the phase noise of the laser in the RF range. This can be then detected by usual devices as ESA. However, it is often difficult to track the beating signal in an electrical spectrum analyzer due to the frequency drift of the laser. To overcome this frequency drift we need more extra efforts to phase-lock two lasers one with each other. The drawback of this method is that low-frequency noise characteristics of the laser under tests are usually lost. When the laser phase noise is lower than the phase noise of the available reference laser, two identical lasers are built and their phases noise compared.

The second method uses a frequency discriminator. Laser frequency fluctuations
are converted into intensity fluctuations by a discriminator like Fabry-Perot etalons
[61, 62, 63], atomic transitions [64] or interferometers [65, 66]. With this method, the
discriminator is operating at a specific frequency and could imply a narrow frequency
bandwidth when it has a high resolution. Therefore an active locking of the laser fre-
quency onto the frequency operating point of the discriminator is usually necessary [67].

The third one is based on the delayed self-heterodyne method, which was first pro-
posed by Okoshi et al. [68] for measuring the linewidth of a laser. Later, this method
was extended to measure the phase noise of a laser by using a short delay [65, 67]. Gen-
erally, these two methods are called uncorrelated and correlated delayed self-heterodyne
respectively. For the uncorrelated delayed self-heterodyne, the delay time is much longer
than the coherence time of the laser, which directly gives only the integrated linewidth of
the laser and it is difficult or even impossible to estimate the intrinsic linewidth. For cor-
related delayed self-heterodyne, the delay time is shorter than the laser coherence-time,
so that the phase of the two fields are statistically dependent. In this case, beating fields
contain information on laser frequency noise. This method is used to characterize the
laser frequency noise in this manuscript.

<figure><img src="image_8.png" /><figcaption>Figure 1.5: Self-heterodyne frequency noise measurement setup. OC: Optical coupler, PC: Polarization controller, AOM: Acousto-optical modulator, PD: photodetector, TIA: Transimpedance amplifier, SSA: Signal source analyzer.</figcaption></figure>

Figure (1.5) is a schematic sketch of the frequency noise experimental setup based on the correlated delayed self-heterodyne technique. The optical signal is separated into two arms by a coupler. One arm is delayed by a fiber length ($L_d$) corresponding to a delay time ($\tau_d$), which is taken shorter than the laser coherence-time ($\tau_c$). In the other arm, the optical frequency is shifted by an acousto-optic modulator ($f_{AO} = 200$ MHz). RF beating of the two fields, detected on a photodetector (PD) is also shifted by $f_{AO}$; it is then amplified using an electrical amplifier (TIA). A polarization controller is placed in one arm in order to maximize the beating signal. The signal source analyzer is used to measure the power spectral density of the phase noise RF beat signal at $f_{AO} = 200$ MHz. In order to calculate the frequency noise, we need to find the transfer function of the setup, considering that the complex electrical field of the laser is:

$$E(t) = E_0 e^{i(2\pi\nu_0 t + \phi(t))} \qquad (1.31)$$

Where $E_0$ is the signal amplitude and we consider that the laser has also a low-intensity noise. $\nu_0$ is the laser frequency. $\phi(t)$ is the signal phase. Then the intensity detected by
the photodetector can be written in the following form:

$$I_d(t) = E_0^2 \left\{ 1 + \cos \left[ 2\pi(\nu_0 + f_{AO})\tau_d + 2\pi f_{AO}t + \Delta\phi(t, \tau_d) \right] \right\} \quad (1.32)$$

Where $\Delta\phi(t, \tau_d) = \phi(t) - \phi(t - \tau_d)$ is the phase difference between the two waves. In this approach, we do not take into account the phase noise introduced by the acousto-optic modulator because the laser phase noise is bigger than the electrical oscillator of the acousto optic modulator. The PSD of the laser frequency noise ($S_ν(f)$) is related to the PSD of the laser phase noise ($S_φ(f)$) at the output of the interferometer, following the expression that includes the transfer function of the delayed interferometer [65]:

$$S_{\nu}(f) = \left( \frac{1}{2\pi\tau_{d}\text{sinc}(\pi f \tau_{d})} \right)^{2} S_{\phi}(f) \quad (1.33)$$

The signal source analyzer gives the PSD of the phase noise. The frequency noise can be calculated by dividing the phase noise PSD by the transfer function of the interferometer. The interferometer acts as a low pass filter. The bandwidth is inversely proportional to the time delay of the interferometer. Figure (1.6a) gives an example of a transfer function of the interferometer with a delay of 200 meters. The cut-off frequency is 1 MHz, which gives a 300 kHz bandwidth. Figure(1.6b) plotted the gain conversion and measurement bandwidth in function of delay length. The gain conversion corresponds to the linear slope conversion of the transfer function. We can increase the measurement bandwidth by decreasing the delay length. When we reduce the length, the gain is reduced (see figure(1.6b)) (the transfer function is a cosine function; decreasing the delay length will increase the slope and thus the phase to intensity conversion ratio). Experimentally a compromise has to be found between these two parameters, that we usually want both as high as possible. The important point is to control that there is enough gain to have a signal above the noise floor of the bench. Noise floor measurement is explained in the next section.

### 1.4.4 Noise floor of the measurement bench

It is important to characterize the noise floor of the bench before any measurement of the laser frequency noise. The noise floor is caused by any noise other than that of the laser. The contribution of the noise floor can be due to mechanical and acoustical vibrations of the delay line fiber spool due to the environment. Also, it can be limited
<figure><img src="image_9.png" /><figcaption>Figure 1.6: (a) Transfer function of an unbalanced interferometer ($\tau_d = 200$ m) (b) Cutoff frequency and gain conversion of the test bench, depending on the inserted delay length.</figcaption></figure>

by the instrumental noise like the noise of the signal source analyzer, the noise from the RF modulator, the noise of the photodetector, and of the RF amplifier. Moreover, it can be limited by the intensity noise of the laser.

<figure><img src="image_10.png" /><figcaption>Figure 1.7: Experimental setup for noise floor measurement. OC: Optical coupler, PC: Polarization controller, AOM: Acousto-optical modulator, PD: photodetector, TIA: Transimpedance amplifier, SSA: Signal source analyzer.</figcaption></figure>

In order to measure the noise floor, we used a method proposed by Llopis et al. [69], which is based on balancing the delay length of the interferometer. Two identical fiber spools are inserted in both arms (see figure (1.7)). Indeed, in this configuration, the contribution of phase noise is not coming from the laser itself but rather from the measurement system and its environment. In this case Eq (1.32) can be written as:

$$I_d(t) = E_0^2 \left\{ 1 + \cos \left[ 2\pi(\nu_0 + f_{AO})\tau_d + 2\pi f_{AO}t + \Delta\phi(t, \tau_d) + \phi_{sys}(t) \right] \right\} \quad (1.34)$$
Where $\phi_{sys}(t)$ is the noise from the measurement system and its environment. By balancing the interferometer, then its delay becomes equal to zero ($\tau_d = 0$). In that case, the phase delay of $\Delta\phi$ is null. As the delay is null, the beating between the lines from the two arms gives a coherent signal only at the same frequency under the laser line, bringing a noise component at DC and negligible contributions at other frequencies. It follows that with a null delay the measured noise is only due to the environment and may be attributed to the interferometer and not to the laser. We used this measurement method to characterize the noise floor before each phase noise measurement.

1.4.5 Example of noise measurement

We have explained the frequency noise and its measurement in the above subsections. In this subsection, we will characterize the frequency noise of a Tunics OM$^{TM}$ laser and estimate its linewidth by using the *beta-line* approximation. As we discussed previously, for each measurement, we should analyze the system noise floor. Figure (1.8a) shows the laser phase noise (blue line) and its interferometer noise floor for different delays. These different delay lengths in MZI enable to improve the gain or the bandwidth. MZI noise floor is given for three delay lengths. First, the phase noise of the modulator signal is plotted in orange color: it is a measure by a direct plug to the SSA. Firstly, we remove the fiber spool from both arms of the MZI, which corresponds to the red curve (0 m) in figure (1.8a). The system sensitivity is measured from other contributions than that of the laser itself. In this case, noise from the fiber spool is not taken into account so it is an optimistic evaluation of the noise floor. The fiber spool may generate noise due to acoustic and mechanical vibrations, and also due to temperature sensitivity of the spool. The identical fiber spools are inserted in each arm to nullify the MZI delay in order to improve the measurement of the noise floor. The additional noise from the previous case is mainly due to the fiber spool. In figure (1.8a), noise floor measurements are shown for 10 and 200 meters of fiber spools in black and pink colors respectively. When the fiber length is increased from 10 meters to 200 meters the phase noise floor of the MZI is increased in the low offset frequency region. For higher frequencies (above 50 kHz), the system noise floor is equal for all cases. Recall that these measurements can be limited by the laser RIN and noise from the photodetector and amplifier. Tunics OM laser phase noise, measured with 10 meter fiber spool delay is plotted in blue color (see figure 1.8a).
<figure><img src="image_11.png" /><figcaption>Figure 1.8: (a) Phase noises of the laser and noise floors of the bench for different fiber lengths. (b) Frequency noise measurement of the Tunics OM laser and its linewidth estimation using *β − line* approximation.</figcaption></figure>

Measured laser phase noise is higher than its noise floor. Also, we can see that the first null at 20 MHz implies a measurement sensitivity that locally drops down to the noise floor.

As we have seen in Eq (1.33), the PSD of the measured laser phase noise is related to the time delay of the MZI. Calculated frequency noises are plotted in figure (1.8b). The gray dotted line is the noise floor with the 10 m-fiber delay. The frequency noise of the Tunics OM laser is plotted (blue curve) and the white and flicker noise contributions are clearly identified. The grey area under the curve is the one used for the estimation of integrated linewidth. Below 1 kHz, laser experiences a plateau in noise, which may be due to the laser driver. Sharp peaks may arise from thermal and mechanical fluctuations. As discussed in section (1.4.2), intrinsic linewidth of the laser may be estimated from the white noise and integrated linewidth can be evaluated using the *β − line* approximation. 6.28 kHz intrinsic linewidth and 203 ± 20 kHz integrated linewidth for an observation time of 10 ms are obtained. In the next section, we are going to explain how to reduce the technical noise and improve the laser-frequency stability.

1.5 Laser-frequency stabilization

Laser frequency stabilization consists in controlling the laser frequency by a servo-control.
This section gives the basic principles of stabilization and the method that is used in this
thesis. The detailed explanation can be found in several references [49, 70, 71]. The idea
<figure><img src="image_12.png" /><figcaption>Figure 1.9: General scheme for frequency stabilization of a laser.</figcaption></figure>

behind the stabilization is to measure the instantaneous laser frequency fluctuations using a frequency discriminator. The common method is to compare the laser frequency with a reference frequency. The frequency difference between the reference frequency and the laser frequency is called "error signal". The reference can be a laser or an optical resonator. The idea is to transfer the stability of the reference to the laser in a similar way that an optical cavity can be stabilized by a reference laser. There are several techniques available to do that, like atomic transitions, optical cavity resonance (Fabry-Perot resonator) or optical interferometers (Fiber Mach-Zehnder). Atomic transitions and interferometers give long-term stability when the environmental or operational conditions are minimized. From this measurement, a correction signal is generated through the actuator. This signal is added to the laser pump in order to ideally compensate frequency fluctuations as a current change generates a frequency shift. The block diagram is given in figure (1.9), with its various elements, and their roles.

Laser-frequency fluctuations are monitored with a discriminator that converts linearly the frequency fluctuations into voltage fluctuations with a conversion slope $K_b$. This discriminator can be a cavity, gas cell or an interferometer. It generates an error signal, which is amplified and compensated by the loop filter. A PID (Proportional, Integrator, Derivative) device is generally used as the loop filter. It has a frequency-dependent gain coefficient $G_0(f)$. The amplified voltage fluctuations feeds the laser through an actuator, which converts voltage fluctuations into frequency fluctuations by a constant $K_a$. This actuator is the laser voltage driver.
Let consider that the initial frequency of the free running laser is $\nu_i$. Assume that it is controlled by a servo loop, which corrects the laser frequency to $\nu$ that is close to the reference frequency $\nu_0$:

$$ \nu = \nu_i - K_a V_c \qquad (1.35) $$

$\nu$ is considered to be the stabilized laser frequency. $V_c$ is loop filter output voltage. The discriminator response is a function of the laser frequency and its output is a voltage. The discriminator voltage $V_d$ is given by

$$ V_d = K_b \delta \nu \qquad (1.36) $$

where $\delta\nu = \nu - \nu_0$ is the residual frequency deviation from the references frequency $\nu_0$ of the stabilized laser. The error signal will be a voltage proportional to $\nu - \nu_0$. The discriminator voltage $V_d$ is amplified by a loop filter, described by a frequency dependent transfer function $G_0(f)$. Then the loop filter output voltage ($V_c$) is applied to the laser through an actuator:

$$ V_c = G_0(f) V_d \qquad (1.37) $$

The loop equation is

$$ \nu = \nu_i - K_a K_b G_0(f) \delta \nu \qquad (1.38) $$

which can be written:

$$ \frac{\delta \nu}{\Delta \nu} = \frac{1}{1 + G(f)} \qquad (1.39) $$

Where $\Delta\nu = \nu_i - \nu_0$ is the initial frequency offset of the free running laser. $G(f) = K_a K_b G_0(f)$ is the gain of the open servo loop. We can see that servo loop reduces the frequency deviation $\Delta\nu$ of the free running laser by the factor of open-loop gain. Thus, the open-loop gain must be large in order to improve frequency stabilization. In general the gain of the amplifier and the servo elements are frequency dependent. For an optimum performances of the servo loop, we need to know the frequency-dependent transfer function of all components [49].

As we have seen in the previous subsections, the laser has mainly two types of frequency noises: white and flicker noise. A frequency discriminator can detect this low-frequency noise fluctuations and suppress them using a feedback control system. Generally, the frequency noise discriminator linearly converts the optical frequency fluctuations into voltage fluctuations. On the other hand, an ultra high finesse optical cavity may be used to stabilize the laser frequency [72, 73, 74] to correct the fast fluctuations and even
reduce intrinsic laser linewidth. Generally, such a stable laser requires highly compli-
cated pressure and temperature stabilization when high performances are expected. In
this thesis, a Brillouin fiber cavity is used with a frequency stabilization technique, which
enables us to reduce the laser linewidth under the Hz level. In this section, we discuss
only the laser stabilization technique, which we used during this work. Brillouin lasing
and its noise analysis will be discussed in chapter 3 and 4.

As we have seen before the performances of the locking is determined by the gain of the loop, which is dependent on the different transfer functions. One of the factors is related to the discriminator slope. We used an optical fiber ring cavity as a frequency discriminator. The instantaneous laser frequency is compared to that of the cavity-resonance frequency, which gives the error signal. The two common methods are available to measure the error signal: the side-of-fringe and modulation methods.

In the side-of-fringe method, the laser frequency is locked on to the half of the trans-
mission power of the cavity resonances. At this point, resonance slope will be maximum
and the transmission power may be considered proportional to the laser frequency devia-
tions. Sensitivity to the frequency deviation measurements can be improved by increasing
the slope of the discriminator with a low loss cavity. This method is easy to implement.
However, this technique has some disadvantages. Firstly, laser amplitude fluctuations
will be coupled to the frequency fluctuations that will increase the noise level. Secondly,
narrow locking bandwidth, small frequency shifts across the cavity transmission lead to a
change in amplitude, and then modify the locking point. Thirdly, side locking will reduce
the amount of light coupled to the cavity. The modulation method can overcome these
problems by locking the laser frequency onto the top of the resonance. The phase shift
of the reflected light is measured through a modulation frequency. The slow frequency
modulation method is named top-of-fringe locking and the fast modulation method is
called the Pound-Drever-Hall technique. These two methods are discussed in the next
section.

**1.5.1 Top-of-fringe locking**

As we have seen, in the side-of-fringe method, the laser is locked onto the side of a
resonance. The intensity measurement gives the sign of frequency deviation. On the
contrary, if we lock onto the top of a resonance, it is not possible to identify the sign
of a frequency change by a direct intensity measurement because in that case, measured voltage decreases in both variations (positive or negative). To overcome this difficulty, slow frequency modulation is applied to the laser output. When the laser frequency is lower than the cavity-resonances frequency, the frequency fluctuations will be converted to amplitude fluctuations with a certain phase. When the laser frequency is greater than the cavity-resonance frequency, it will create the same conversion with an opposite phase change. At the resonance frequency, the conversion is equal to zero. This correction gives the first derivative of the cavity transmission line-shape function. From the phase change, we can obtain the sign of the laser frequency fluctuations. This error signal acts as a discriminator; it allows us to lock the laser onto the maximum of the resonance independently of the intensity fluctuations.

<figure><img src="image_13.png" /><figcaption>Figure 1.10: (a) General scheme of top-of-fringe frequency stabilization. (b) Oscilloscope traces for transmission laser signal, modulation and top of fringe error signals. PD: photodetector, PID: Proportional–integral–derivative controller, VC: Variable coupler, FN: Frequency noise measurement bench.</figcaption></figure>

Figure (1.10) shows an experimental sketch of the top-of-fringe laser-stabilization. The fiber ring resonator is used as a discriminator and the Tunics OM laser is locked onto the ring cavity. The laser is modulated at lower frequency compared to the linewidth of the cavity resonance and the transmitter beating signal is detected on the photodetector and is demodulated with the modulation frequency thanks to a mixer. A phase shifter will ensure the input of the mixer to have the proper phase relationship between modulation and detected signal. This modulation and demodulation are done with the help of a locking amplifier. The error signal amplifier and the locking point are controlled by a
PID. The PID output is fed to the laser driver, which corrects the laser frequency. The error signal is shown in figure (1.10b), which has an asymmetry due to the mismatching phase shifter.

To summarize, the top-of-fringe method is independent of the laser intensity fluctuations. One major disadvantage is the correction bandwidth that is limited by the modulation frequency, which has to be smaller than the linewidth of the resonance. It's why this method is difficult to implement on very narrow resonance. Similarly, we can lock the cavity resonance onto the laser frequency, in this case, we modulate and correct the cavity length with a PZT. We used this method for the RIN studies of Brillouin fiber laser. The fast modulation method can overcome the disadvantage of the top-of-fringe method.

1.5.2 Pound-Drever-Hall technique

The fast frequency modulation method called Pound-Drever-Hall (PDH) technique was first described by Ronald Drever and John Hall, based on a frequency-modulation technique for microwave cavities developed by Robert Pound [62]. This part gives a brief introduction to the experimental setup of PDH. A detailed explanation of the PDH method can be found in several papers [75, 76].

<figure><img src="image_14.png" /><figcaption>Figure 1.11: (a) General scheme of PDH frequency stabilization. (b) Experimental Oscilloscope traces for PDH error signal. PM: Phase modulator, PD: photodetector, PID: Proportional–integral–derivative controller, VC: Variable coupler, FN: Frequency noise measurement bench.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 1.12: (a) Frequency noise measurement of the stabilized Tunics laser on fiber ring cavity with top of fringe and PDH lock.</figcaption></figure>

Figure (1.11) shows the experimental setup of the PDH locking scheme. The fiber ring cavity is used as a frequency discriminator. The experimental setup is similar to the top-of-fringe lock. The phase modulator modulates the laser frequency at a modulation frequency that is higher than the cavity resonances linewidth. This phase modulator produces two frequency modulation sidebands equally spaced from the laser carrier. Since they are well outside the cavity resonances, sidebands are fully reflected from the cavity with no significant phase shift. When the laser carrier frequency matches the cavity resonance, then a reflected component from the cavity will be in phase with the incident light. The beating between the carrier frequency and its side-band is detected at a modulation frequency using a photodetector. At resonance, a sideband frequency beating will be $180^\circ$ out of phase with respect to the other one. It follows that the sum of the two beatings at the modulation frequency (that we will call PDH beating in the following) will be canceled out and there will be no signal on the photodetector at this modulation frequency. When the laser frequency is detuned slightly from the cavity resonance, the reflected light at the carrier frequency experiences a phase change; the sign of this phase change is given by the sign of the deviation of the PDH beating. The PDH beating will not be zero anymore because it is not exactly $180^\circ$ out of phase. The PDH beating gives the error signal, which is used to correct the laser frequency (see figure(1.11)). Using a low-pass filter to isolate the DC term from the mixer, the PID output provides the error signal that is applied to the actuator for laser-frequency correction, after appropriate conditioning. During this work, the Toptica PDH controller (PDD 110/F) was used for laser stabilization.
Using this frequency stabilization method the laser frequency will be locked onto a
cavity resonance. A cavity resonance will be generally more stable in the low-frequency
range than a laser frequency. By doing the locking, the cavity stability is transferred
to the laser frequency, which reduces the laser flicker noise. Figures (1.12a) shows the
laser frequency noise measurement for two different locking methods. The blue line gives
the free-running-laser frequency-noise. The black line shows the laser frequency noise
using top-of-fringe locking. The locking improves the frequency noise in the low Fourier
frequencies; the correction bandwidth is limited to 8 kHz. The top-of-fringe method
enables to improve the laser linewidth from 203 ± 20 kHz (for the free-running laser)
down to 72 ± 7 kHz with an observation time of 10 ms. The cyan line gives the laser
frequency noise using a PDH locking; the correction bandwidth is then increased to 50
kHz; the integrated linewidth is reduced down to a value lower than 2 kHz with an
observation time of 10 ms. As the noise floor is reached, the estimated linewidth is an
approximated value, which is overestimated. Moreover, we measure the noise floor of the
bench as shown in figure (1.12b). The PDH locking measurement is limited to the noise
floor of the bench.

**1.6 Conclusion**

In this chapter, basic notions on laser intensity and frequency noise are introduced. Laser
RIN is explained as well as the method of measurement using the direct detection method.
A description of laser phase and frequency noise is detailed as well as the *β* — *line*
approximation method. Self-heterodyne method is introduced for measuring the laser
frequency noise and its noise floor. Moreover, we show that the laser frequency can be
stabilized on a fiber ring cavity resonance using the top-of-fringe and PDH methods.
These servo-loop method will be used to realize Brillouin lasers in the coming chapters.
Before explaining the Brillouin laser under study, we will introduce some generalities
about the stimulated Brillouin scattering and its characterization method in the next
chapter.
<figure><img src="image_16.png" /></figure>
# Chapter 2

## Brillouin gain characterization using
cavity ring-down spectroscopy

### 2.1 Introduction

This chapter is dedicated to a comprehensive overview of Brillouin scattering and detailed discussions about its characterization in optical fibers using different methods. A new method to characterize the Brillouin gain in high-Q cavities by using Cavity-Ring Down Method (CRDM) is proposed and analyzed.

### 2.2 Generalities on stimulated Brillouin scattering

Density fluctuations, impurities and thermal agitations of particles in a medium are among the various sources at the origin of different types of scattering. They are classified as linear, when the optical properties of the medium are not modified by incident light, on the contrary, nonlinear scattering occurs when the light intensity is sufficiently high to modify the optical response of the medium. Among nonlinear scattering effects, we can mention electrostriction in Stimulated Brillouin Scattering (SBS) and molecular vibrational and rotational states in Stimulated Raman scattering (SRS) [77]. Indeed, scattering could be divided into two categories, elastic or inelastic scattering. The energy of an incident photon is conserved in elastic scattering; it's why no frequency shift is observed in the scattered photon. Rayleigh scattering is in this category. On the contrary, for inelastic scatterings, energy exchange between scattered photons and incident
photons leads to a change in frequency as for Raman and Brillouin scatterings.

In this thesis, we focus on spontaneous Brillouin scattering and stimulated Brillouin scattering, which are the object of the next sections. The aim of this section is to introduce generalities on Brillouin scatterings such as standard coefficients and equations that will be largely used in the rest of the thesis. Spontaneous and stimulated Brillouin scatterings are described respectively in sections (2.2.1) and (2.2.2). Some of the explanations and equations are taken from the reference [78]. One objective of this thesis is to design a low-noise Brillouin laser. Gain and threshold of Brillouin scattering are key ingredients to implement those low-noise lasers. We then introduce those parameters in section (2.2.3) and (2.2.4) respectively.

## 2.2.1 Spontaneous Brillouin scattering

In a scattering medium, molecules make small displacements around their position of equilibrium under the effect of temperature. These movements are coupled together and create very small acoustic waves, which propagates in all directions. These acoustic waves modify the local matter-density and cause modifications of the optical index. It creates an index grating that moves at the speed of the acoustic wave. The incident light, for an acoustic wave moving in the same direction, is scattered on this grating with a down-shifted frequency due to the Doppler effect, generating a diffracted wave propagating in the opposite direction. The resulting wave is called a Stokes wave.

When the acoustic wave has an opposite direction to the incident wave, the scattered wave absorbs the energy of phonons and generates photons at a higher frequency called anti-Stokes. This process is called spontaneous Brillouin scattering and it is generated from a thermally excited acoustic vibrations [78]. At room temperature, these acoustic vibrations are present in fiber. Figure (2.1) illustrates spontaneous Brillouin scattering process in terms of wavevector and frequency. Applying the laws of momentum and energy conservations, we can write

$$ \vec{k}_S = \vec{k}_P - \vec{q} \qquad (2.1) $$

$$ \omega_S = \omega_P - \Omega \qquad (2.2) $$

Here $\vec{k}_i$, $\omega_i$ ($i = P, S$) are the wavevector and angular frequency of pump and Stokes waves respectively. $\vec{q}$, $\Omega$ are wavevector and angular frequency of the acoustic wave,
<figure><img src="image_17.png" /><figcaption>Figure 2.1: Schematic representation of Brillouin scattering process and schematics diagram of (a) the propagation direction and (b) Momentum conservation of the incident, scattered (Stokes in orange) and acoustic wave. Red dashed lines represent anti-Stokes process.</figcaption></figure>

respectively. Frequencies and wavevectors are related by

$$|\vec{k}_p| = \frac{2\pi n}{\lambda_p}, \quad |\vec{k}_s| = \frac{2\pi n}{\lambda_s}, \quad |\vec{q}| = \frac{\Omega}{V_A} \qquad (2.3)$$

Here $n$ is the optical refractive index of the medium, and $V_A$ is the longitudinal acoustic velocity. As illustrated in figure (2.1) the conservation of momentum with an angle $\theta$ between pump and Stokes wavevector writes as follows:

$$|\vec{q}|^2 = |\vec{k}_p|^2 + |\vec{k}_s|^2 - 2|\vec{k}_p||\vec{k}_s| \cos \theta \qquad (2.4)$$

Since $|\vec{k}_p|$ is nearly equal to $|\vec{k}_s|$ ($\Omega \ll \omega$)

$$|\vec{q}| = 2|\vec{k}_p| \sin \frac{\theta}{2} \qquad (2.5)$$

Substitution of Eq (2.5) into acoustic angular frequency shift relation (2.3) gives:

$$\Omega = \frac{4\pi n V_A}{\lambda_p} \sin \frac{\theta}{2} \qquad (2.6)$$

In single-mode optical fibers, the only permissible angles are $\theta = 0$ or $\theta = \pi$. $\Omega$ is equal to zero for forward scattering and is maximum for back-scattering. This gives the Brillouin frequency shift $\nu_B$ ($\Omega_B = 2\pi\nu_B$ at $\theta = \pi$):

$$\nu_B = \frac{2nV_A}{\lambda_p} \qquad (2.7)$$

From this, we can understand that the Brillouin frequency shift depends on the pump wavelength and material properties through the acoustic velocity and the refractive index. Similarly, the anti-Stokes component with up-shifted frequency can be obtained. When
<figure><img src="image_18.png" /><figcaption>Figure 2.2: Stokes process in (a) the spontaneous regime. Amplitudes of Stokes and anti-Stokes are of same order. (b) Stimulated Brillouin scattering regime. Stokes amplitude will increase and its gain spectrum narrows. $\nu_B$: Brillouin shift, $\nu_P$: pump frequency, $\Delta\nu_B$: Brillouin gain bandwidth and $g_B$: Brillouin gain coefficient</figcaption></figure>

acoustics wave is counter-propagating to the pump wave (see figure 2.1a red line), we obtain:

$$ \bar{k}_{AS} = \bar{k}_P + \bar{q} \qquad (2.8) $$

$$ \omega_{AS} = \omega_P + \Omega \qquad (2.9) $$

Where $\bar{k}_i$, $\omega_i$ ($i = P, AS$) are, as mentioned above for the Stokes, respectively the wavevector and angular frequency of the pump and of the anti-Stokes waves. In Stokes and anti-Stokes processes, the incident wave interacts with different phonon ensembles. For Stokes process, the pump wave interacts in the same direction as the acoustics wave. For anti-Stokes process, acoustics waves propagate in the opposite direction to that of the pump. In spontaneous Brillouin scattering, the Stokes and anti-Stokes wave have almost the same amplitude because the sources of acoustic phonons are only due to thermal agitation. Figure (2.2a) shows the spontaneous Brillouin scattering spectrum of Stokes and anti-Stokes. The spectral broadening of the two waves are due to the damping of acoustic wave; it is called spontaneous Brillouin gain bandwidth ($\Delta\nu_B$) and it is a unique feature of the Brillouin process. When the injected pump power is increased, the amplitude of the acoustic waves increases leading to efficient Stokes wave generation. This phenomenon is called stimulated Brillouin scattering and is described in the section below.

### 2.2.2 Stimulated Brillouin scattering

As we have seen previously, spontaneous Brillouin scattering is the result of the interaction between a pump wave and acoustic phonons induced by thermal agitation. The efficiency
of these spontaneous scattering are very low compared to stimulated process. Pump and Stokes waves beat together in the fiber giving rise to an interference pattern oscillating at the Stokes frequency shift $\nu_B$ (figure 2.3). For higher pump power, the amplitude of this beating increases and implies a modification of the matter density along the spatial pattern. This physical phenomenon, called electrostriction, is the capacity of a material to be compressed in the presence of an electric field. This electrostriction process induces a moving index grating at the speed of sound ($V_A$) in the medium, which selects a set of frequencies ($\omega_S, \Omega_B$) for a given pump frequency ($\omega_P$). This means that electrostriction process creates a large number of acoustic phonons, which will tend to amplify the Stokes wave generation. The created Stokes wave and the pump wave interferes. This interference itself creates a material grating that increases the beating strength, which reinforces the Stokes wave generation (figure 2.3). Finally these two processes reinforce each other and causes the amplification of Stokes wave. This looping process is called stimulated Brillouin scattering.

<figure><img src="image_19.png" /><figcaption>Figure 2.3: Schematic representation of the SBS. The beat between the pump wave and the Stokes wave creates by electrostriction and acoustic wave that diffuses the pump wave and therefore reinforces the Stokes wave.</figcaption></figure>

It is interesting to note that due to the energy conservation condition, scattering Stokes process generates phonons. On the opposite, for anti-Stokes process, the phonons are absorbed by the pump wave in order to generate anti-Stokes photons at higher frequency. In spontaneous regime, Stokes and anti-Stokes have an equal probability to appear and then show the same amplitude (figure 2.2a). In the stimulated regime, high pump power
and electrostriction process rapidly increase the number of phonons, which propagate in the same direction as the one of the pump; it then amplifies the Stokes wave. In contrast, the anti-Stokes process is generated from the annihilation of pump photons and acoustic phonons, which propagate in opposite directions. These annihilations are a strong limitation to the amplification of anti-Stokes (figure 2.2b).

In conclusion, back-scattered Stokes wave is created by spontaneous Brillouin scattering from thermal noise. In stimulated Brillouin scattering, this Stokes wave is amplified through the electrostriction process. This amplification process may be used for the realization of low-noise Brillouin laser. The gain and threshold of Brillouin process are important to study for the design of laser. These two quantities are described in next sections.

### 2.2.3 Brillouin gain

As we have seen previously, Stokes wave amplitude will grow very rapidly in the stimulated Brillouin scattering regime. This growth will be characterized by the Brillouin gain $g_B(\nu)$. The spontaneous Brillouin gain spectrum is given by:

$$g_B(\nu) = g_B \frac{(\Delta\nu_B/2)^2}{[(\Delta\nu_B/2)^2 + (\nu - \nu_B)^2]} \quad (2.10)$$

The Brillouin gain is maximum at the frequency of $\nu = \nu_B$ and is equal to the Brillouin coefficient:

$$g_B = \frac{2\pi n^7 p_{12}^2}{c \rho_o \lambda_p^2 \Delta\nu_B V_A} \quad (2.11)$$

Where $p_{12}$ is the elasto-optic constant and $\rho_0$ is the matter density. The Brillouin gain bandwidth is $\Delta\nu_B$ related to the acoustic phonon lifetime ($\tau_a = 1/2\pi\Delta\nu_B$) and to the acoustics absorption coefficient ($\alpha_s = \Delta\nu_B/V_A$). Generally, for standard telecommunication optical fiber, the Brillouin gain bandwidth $\Delta\nu_B$ is 30 MHz, which corresponds to a phonon lifetime of 5.3 ns. The acoustic waves propagate on short distances compared to the optical wave. Typically, Brillouin gain coefficient of standard silica fiber are ranging from $1.5 \times 10^{11}$ to $3 \times 10^{11}$ m/W [79, 80]. It is 1000 times higher than the Raman gain coefficient in silica fiber [78]. In fact, the Brillouin gain coefficient strongly depends on the composition of the medium; for example, chalcogenide fiber may have 100 times higher Brillouin gain coefficient than silica fiber [23].

Stimulated Brillouin scattering can be used as an amplification process in fiber. Two configurations may be considered. Firstly, the generator configuration, for which the
<figure><img src="image_20.png" /><figcaption>Figure 2.4: Schematics of the pump-probe method. $P_p$ and $P_s$ are the pump and Stokes powers respectively.</figcaption></figure>

pump is injected into the fiber and the thermal noise initiates the Stokes wave. Secondly, the amplifier configuration, for which the pump is injected from one side and a weak signal (called probe) from the other side (see figure 2.4). If the probe frequency satisfies the Brillouin conditions (Eq. 2.2), then the probe signal will be amplified. Hence, the fiber acts as a gain medium. The amplification bandwidth is set by the acoustic phonon decay rate in the medium, which is related to the Brillouin gain bandwidth.

In the following, we will estimate the Brillouin single pass gain (G). To get a first estimation, we neglect the depletion of the pump intensity, by the nonlinear process. Then intensity evolves as $I_P(L) = I_p(0)e^{-\alpha_f L}$, where $I_p(0)$ is the intensity of the pump injected at $z = 0$. $L$, $\alpha_f$ are the fiber length and the losses coefficient respectively. It follows that the Stokes intensity along the fiber will vary as [78]:

$$ \frac{dI_S}{dz} = -k_{\text{polar}} g_B I_p(0) e^{-\alpha_f z} I_S + \alpha_f I_S \quad (2.12) $$

Where $k_{\text{polar}}$ is a polarization factor, that takes into account the variation of the polarization between pump and Stokes waves. $k_{\text{polar}}$ varies between 0.5 and 1 depending on the birefringence and on the length of the fiber [81, 82]. A solution can be found to this equation:

$$ I_S(0) = I_S(L) e^{k_{\text{polar}} g_B \frac{P_P(0)}{A_{\text{eff}}} L_{\text{eff}} - \alpha_f L} \quad (2.13) $$
Where, $A_{\text{eff}}$ is the effective mode area, $L_{\text{eff}} = \frac{1}{\alpha_f} \left[ 1 - e^{-\alpha_f L} \right]$ is the effective length.

The total gain of this optical system may be defined as the ratio of the input and output Stokes intensity $I_S$ (considering that the Stokes wave is contra-propagative from z=L to z=0):

$$G = \frac{I_S(0)}{I_S(L)} = \frac{P_S(0)}{P_S(L)} = e^{k_{\text{polar}} g_B \frac{P_S(0)}{A_{\text{eff}}} L_{\text{eff}} - \alpha_f L} \quad (2.14)$$

where G is equal to the net gain of the system. This relation is used for gain characterization based on pump-probe experiments. In this thesis, we used this relation to analyse our experimental observations obtained from the CRDM technique (Section 2.4). As we have seen before the stimulated Brillouin scattering exponentially increases in higher pumping power regime. In the next section, we are going to determine the critical power corresponding to the stimulated Brillouin scattering threshold.

### 2.2.4 Brillouin threshold

In stimulated Brillouin scattering process, most of the pump power is transferred to the Stokes wave at a critical pump intensity, which is called Brillouin threshold $P_{th}$. Above this threshold, the Stokes intensity exponentially increases in the opposite direction of the pump. In this section, we determine the threshold power of stimulated Brillouin scattering. The Brillouin threshold is defined variously in the literature but all definitions share the same concept [83, 84, 85].

<figure><img src="image_21.png" /><figcaption>Figure 2.5: Definition of Brillouin threshold power using transmitted (red) and back scattered Stokes (black) signals plotted as a function of the input power. 1 % of the input signal is plotted in blue color for references. $P_p(0)$: input pump power, $P_S(0)$: back-scattered Stokes, $P_p(L)$: transmitted power; (a) Logarithmic scale (b) Linear scale.</figcaption></figure>
In figure (2.5), back-scattered Stokes ($P_s(0)$) (black curve) and transmitted power ($P_p(L)$) (red curve) are drawn as a function of the input power ($P_p(0)$). Figure (2.5 a) is plotted in logarithmic scale. It clearly highlights the Stokes transition from spontaneous to stimulated Brillouin scattering (black line). The transmitted power linearly increases in the spontaneous regime and then starts to saturate, indicating the onset of the stimulated regime, at a power, that can be defined as the threshold $P_{th}$. The Stokes and transmitted powers are intersecting at one specific point, that could be also taken as the threshold. 1 % of input power is plotted in blue, which is intersecting the Stokes power (black curve) at the center of the transition in logarithmic scale. Figure (2.5 b) shows the same figure plotted in linear scale. Taking 1 % of input power curve is of interests as it corresponds to an intermediate position in the transition between the end of the spontaneous regime (the first linear progression) and the starting of the stimulated one (the second linear dependence at higher pump powers). More generally, the threshold is defined as the input pump power, which is necessary to get a, back-scattered power equal to a fraction $\mu$ of the pump power: $P_s(0) = \mu P_p(0)$. $\mu$ may be taken equal to 0.01 [83, 84]. The idea behind this approach is that near threshold the Stokes power follows an exponential behavior and starts to approach an amplitude close to the one of the input power (2.5 b). An analytical equation for threshold can be found in [34, 86]:

$$P_{th} = \frac{g_{th}A_{eff}}{k_{polar}g_{B}L_{eff}} \qquad (2.15)$$

Where $g_{th}$ is a factor, which is dependent on $\mu$ and on material parameters (see appendix B). This dependency implies that the determination of the threshold from Eq (2.15) involves the knowledge of material parameters of the fiber under study, which are not always available.

In conclusion, Brillouin scattering is the interaction between three waves, which are Stokes, pump and acoustic waves. In spontaneous Brillouin scattering, Stokes wave are generated from interaction of pump wave and a thermally excited acoustic wave. In stimulated Brillouin scattering, the excited acoustic wave and Stokes wave are amplified by the electrostriction process in presence of higher pumping powers. In this thesis we are focused on the development of low-noise Brillouin laser. Indeed, it is important for us to measure the SBS gain bandwidth and gain coefficients to improve the laser design.
In this chapter, we propose an intracavity gain characterization method based on cavity ring-down method (CRDM). In order to compare and validate our experimental method, we characterized the Brillouin gain coefficient through conventional method that uses single pass Brillouin gain determination (section (2.3)). In section (2.4), we explain CRDM and its results. In section (2.4.7), we compare gain coefficients values extracted from CRDM, conventional method and results from the literature in order to validate our new method that allows Brillouin gain to be characterized directly in a laser configuration.

## 2.3 Experimental characterization of Brillouin gain

Stimulated Brillouin scattering in optical fiber can be experimentally characterized by analyzing the Brillouin scattering spectrum. As previously mentioned, the frequency of back-scattered light is shifted from its pump frequency. Shifted Brillouin spectrum experiences a broadening due to the acoustic waves usually following a Lorentzian shape [85]. At a given pump wavelength, the Brillouin spectra (Eq. 2.10) is defined by three parameters: the Brillouin shift ($\nu_B$), the Brillouin gain bandwidth ($\Delta\nu_B$) and the Brillouin gain coefficient ($g_B$). These parameters (see table 2.4) are strongly dependent on fabrication process for conventional silica fibers (for example Corning SMF-28) [5, 33, 87, 88]. Also some experimental studies have been made in microstructured fiber, the coupling between optical and acoustical waves may be modified. It may have an impact on the three parameters that govern Brillouin backscattering and thus on the Brillouin threshold of the fiber as demonstrated in silica microstructured fibers [84, 89]. Similarly, Brillouin characterization of chalcogenide glass fibers [23, 90, 91] and planar wave guide [21] were reported. Chalcogenide glasses have higher Brillouin gain coefficients than SMF-28 due to material properties. Such values are reported in Kenny Hey Tow thesis [33], for microstructured chalcogenide fibers, in which the lower effective modal area enhances the nonlinear effect.

In 1972, Ippen et al. [11] have proposed the first measurement method based on Fabry-Perot etalon, used to determine the Brillouin shift $\nu_B$ and gain bandwidth $\Delta\nu_B$, by analyzing back-scattered wave of Brillouin scattering from optical fibers [11]. However the spectral resolution was not good enough to measure precisely the Brillouin gain bandwidth. Shibata et al. [92] proposed a pump-probe method to determine the on-off gain(G)$^1$ and

<footnote>$^1$on-off gain: Stokes-power difference with and without Brillouin gain Eq.(2.14)</footnote>
Brillouin gain bandwidth. Two laser sources are required: a pump and a probe, for which signals are contra-directional. The probe frequency, which is shifted from the pump by the Brillouin shift, is tuned finely to scan the Brillouin gain line. Nikles et al. [80] used a single laser source. They generate pump and probe signals using electro-optic modulator. Another method to measure the Brillouin gain spectrum is the self-heterodyne technique, which was proposed by Tkach et al. [93]. In this method, the beating between the pump and the back-scattered wave is simply analyzed with an RF spectrum analyzer.

The above mentioned methods require input powers of the order of hundreds of milliwatts to watts or fiber length of the order of 10's of km, so that they cannot be used in case of short lengths (integrated resonators or micro-resonators) or low-power damage-threshold photonic structures. Moreover the techniques listed above do not allow to characterize the gain inside a resonator configuration.

To overcome these drawbacks, we propose in the following to adapt an existing technique, based on cavity ring-down method (CRDM), to the Brillouin gain characterization inside a high-Q resonator. Before describing the CRDM technique, we determine the Brillouin gain coefficient using conventional methods. In this section, two Brillouin gain characterization methods are analyzed and compared using the same fiber. In section (2.3.1) we describe the self-heterodyne method, which is based on measuring the back-scattered Stokes frequency and the gain bandwidth. From Brillouin gain bandwidth, we can estimate the Brillouin gain coefficient using Eq. (2.11) by knowing the parameters. In section (2.3.2), we verify the Brillouin threshold using estimated Brillouin gain coefficient from the self-heterodyne technique.

### 2.3.1 Self-heterodyne technique

In the following, we describe the self-heterodyne (SH) technique proposed by Tkach et al. [93] and used for the Brillouin gain characterization in various media as silica optical fibers [93], $As_2Se_3$ chalcogenide fibers [23], tellurite fibers [94], Bismuth-oxide fibers [95], polymer optical fibers [96] and photonic crystal fibers [84, 89]. Among the various approaches listed above to measure the Brillouin gain, we described in more detailed the self-heterodyne technique since this is the one usually used in our lab.

We will then compare the gain evaluation obtained by both the SH technique and the CRDM on a given fiber. The objective is to calibrate the CRDM technique and to
confirm its ability to be used for Brillouin gain determination in high-Q resonators.

<figure><img src="image_22.png" /><figcaption>Figure 2.6: Experimental setup of self-heterodyne technique for measuring the Brillouin gain bandwidth. EDFA: Erbium doped fiber amplifier, VOA: Variable optical attenuator, PM: Powermeter, FUT: Fiber under test (L=26 km), PC: Polarization controller, PD: Photodetector, ESA: Electrical Spectrum Analyzer, OC: Optical coupler.</figcaption></figure>

The self-heterodyne method is based on the measurement of Brillouin gain spectrum by beating pump and back-scattered Stokes waves. The beating signal is measured using a fast-photodetector and a RF spectrum analyzer. Therefore, the beating between the pump and Stokes waves, separated from the Brillouin shift, will directly give the gain spectrum; the laser pump is acting as a local oscillator when pump linewidth is much lower than that of the Brillouin gain spectrum.

Figure (2.6) gives a schematic of the experimental setup for the self-heterodyne technique. The pump laser is a narrow linewidth fiber laser (from Koheras), with less than 2 kHz linewidth, emitting at 1.55 µm wavelength. The power emitted by the pump laser is splitted into two arms, by a 50/50 coupler (OC1). The first arm is called the "pump" arm and the second one the "reference" arm. The signal from the "pump" arm is amplified by an EDFA and then controlled by a variable optical attenuator (VOA). 1 % of the pump is extracted through a coupler (OC2) for power monitoring before being injected into the fiber under study via an optical circulator. The Stokes wave is mixed with the signal of the "reference" arm by using a 50/50 coupler (OC3). The resulting beat signal is transformed into an electrical signal by a fast-photodetector (20 GHz bandwidth), amplified and analyzed by an Electrical Spectrum Analyzer (ESA). Note that a polarization controller (PC) is inserted into one of the two arms of the coupler to maximize the amplitude of the beat signal.
<figure><img src="image_23.png" /><figcaption>Figure 2.7: (a) Spontaneous Brillouin spectrum at -6.8 dBm with Lorentzian fit. (b) Stimulated Brillouin spectrum at 9.2 dBm with Gaussian fit.</figcaption></figure>

As we have seen before in section (2.2.4), Brillouin threshold is directly related to the fiber length so we used a 26 km long SMF-28 optical fiber for our Brillouin gain characterizations. Brillouin scattered spectra for different pump powers are shown in figure (2.7 a, b). The beating signal between Stokes wave and reference laser are centered at 10.87 GHz, which corresponds to the Brillouin shift of SMF-28 fiber. The FWHM of the scattering spectrum can be extracted from these experimental data by superimposing the experimental curves either with a Lorentzian or a Gaussian curve. Doing so, the profile of the scattering spectrum is obtained and FWHM is computed analytically. Figure (2.7a) shows the experimental gain curve and the associated Lorentzian fit at low power. As the pump power is increased, the back-scattered signal linewidth becomes narrower as reported in Gaeta *et al.* [85] and it can be approximated by a Gaussian fit.

In spontaneous regime, back-scattered Stokes linewidth depends only on the acoustic phonons decay rate, which is the spontaneous Brillouin gain spectrum. In stimulated regime, the back-scattered light will be amplified by the electrostriction process that will narrow the back-scattered Stokes linewidth. The evolution of the FWHM of the Brillouin spectrum as a function of the power injected into the silica fiber is plotted in figure (2.8). We can see that the FWHM for spontaneous regime is 27.5±2 MHz. This value decreases exponentially when the injected power increases; finally it is saturating around 6.5 ± 2 MHz. The Brillouin gain bandwidth can be extracted from the spontaneous Brillouin spectrum which gives $\Delta\nu_B = 27.5$MHz.

As we have seen in section (2.2.3), we can estimate the Brillouin gain coefficient from the spontaneous Brillouin gain bandwidth and material parameters of the SMF-28 fiber.
<figure><img src="image_24.png" /><figcaption>Figure 2.8: FWHM of Brillouin spectra as a function of the input pump power in SMF-28 fiber.</figcaption></figure>

We recall the Brillouin gain coefficient Eq. (2.11)

$$g_B = \frac{2\pi n^7 p_{12}^2}{c \rho_o \lambda_p^2 \Delta \nu_B V_A} \qquad (2.16)$$

Where, material parameters of the SMF-28 are given in table 2.3 and 2.4. The obtained Brillouin gain coefficient $g_B = 1.92 \pm .14 \times 10^{-11} m/W$ is laying within a range of reported values $1.9 - 2.5 \times 10^{-11} m/W$ in the literature [79, 80].

### 2.3.2 SBS threshold

In this section, we want to verify the experimental estimation of $g_B$ performed in the section above using the SH technique. To do so, we will proceed using the following steps. First we will analytically estimate the Brillouin threshold from Eq. (2.11) using the previous experimental estimation of $g_B$. Then, we will compare the estimated $P_{th}$ value to an experimental characterization of $P_{th}$ using the definition given in section (2.2.4). We will conclude by a discussion on the matching between the estimated $P_{th}$ and the experimentally determined value. As we have seen in section (2.2.4), the Brillouin threshold is proportional to $g_{th}$ through material parameters (see Eq. 2.15). We substitute the Brillouin gain coefficient from self-heterodyne and other parameters (see table 2.3) in Eqs. (B.2-B.5) and estimate that $g_{th}$ is equal to 11.24 for 26 km long fiber with losses of 0.2 dB/km. Similarly, SBS threshold was calculated from Eq. (2.15) taking into account the following parameters $A_{eff} = 84.90 \mu m^2$, $k_{polar} = 2/3$, $L = 26$ km ($L_{eff} = 15$ km) and $g_B = 1.92 \pm .14 \times 10^{-11} m/W$. We have an estimated SBS threshold of $P_{th} = 4.90 \pm .80 mW$.
<figure><img src="image_25.png" /><figcaption>Figure 2.9: Experimental setup for SBS threshold measurement. EDFA: Erbium doped fiber amplifier, VOA: Variable optical attenuator, FUT: Fiber under test (L=26 km), PM: Powermeter, OC: Optical coupler.</figcaption></figure>

This value will be compared in the next paragraph to the Brillouin threshold estimation
through L-I curve measurement.

### Brillouin threshold estimation through L-I curve measurement

The experimental estimation of the SBS threshold of the optical fiber requires the mea-
surement of transmitted and back-scattered powers according to the input pump power
into the fiber as shown in figure (2.9). The pump laser is a DFB-fiber laser from Koheras,
emitting at 1.55 µm wavelength. The laser output is amplified using an Erbium-doped
fiber optical amplifier (EDFA) before being injected into the optical fiber under test
(FUT) via port 2 of an optical circulator. A variable attenuator (VOA) and a 99/1 cou-
pler were placed before port 1 of the circulator to vary and measure the injected pump
power (P<sub>in</sub>) into the FUT. The back-scattered power (P<sub>S1</sub>) is read using a power-meter
placed on port 3 of the circulator. Transmitted power (P<sub>trans</sub>) is measured at the end
of the FUT. In figure (2.10) are shown the back-scattered and transmitted powers as a
function of the injected pump power. The straight line corresponds to 1 % of the injected
pump power. We can determined from figure (2.10), the Brillouin threshold identified as
the transition from spontaneous to stimulated regimes. 1 % of the injected pump power
is intersecting the Stokes signal at the center of the S shape curve. At the same pump
power, the transmitted pump signal started to experience a saturation due to the non-
linear process that efficiently transfer pump power towards the Stokes. Then Brillouin threshold of 4.78 mW (6.8 dBm) was measured, which is in agreement with the estimated value $P_{th} = 4.90 \pm .80$ mW obtained from SH technique.

<figure><img src="image_26.png" /><figcaption>Figure 2.10: Brillouin threshold estimation using L-I curve measurement for a 26 km long SMF-28.</figcaption></figure>

In the above sections, we have explained the determination of Brillouin gain coefficient using a self-heterodyne technique. We compare the Brillouin threshold estimations by analytical and experimental methods. This comparison validates our estimated Brillouin gain coefficient. These methods require a balance between the fiber length and the injected pump power. In case of short length, as in integrated waveguide and micro-ring cavities, the injected pump power becomes too important to adopt these methods especially because the interaction length is short (larger level of losses with respect to the fiber). It is why in the next section we introduce a new gain characterization method based on cavity ring-down method (CRDM), that requires less fiber length and injected pump-power because the re-circulation of light inside the resonator will increase the effective length of interaction.

## 2.4 Cavity ring-down method

The cavity ring-down method (CRDM) is a well-known technique in optical spectroscopic that measures the decay rate of light injected in an optical resonator. In 2008, Institute FOTON published that CRDM can be used to determine the coupling regime for passive resonators and the resonant gain for active resonators. The method was tested on $Er^{3+}$ doped fiber resonators and also applied to determine the intrinsic and external Q-factors
of an $MgF_2$ whispering gallery mode resonator [97]. Later, it has been implemented on Erbium doped fluoride glass WGM microresonator [2], and studied the possibility of controlling the coupling regime of an ultrahigh Q-factor microresonator from undercoupling to spectral selective amplification by changing the pumping rate. Also, it was shown that transmission and dispersive properties of the microresonators can be characterized using the CRDM. In 2010, Trebaol et al. [98], have shown that the CRDM technique can accurately measure linear properties and frequency splittings of high-Q factor WGM microresonators. In 2015, CRDM was extended to the nonlinear regime study in [99], which enables the input power and the mode volume to be estimated for microspheres. Savchenkov et al. [100] reported an experimental observations of power-dependent and nonexponential decay of light stored in whispering gallery modes caused by stimulated Raman scattering in the resonator using cavity ring-down method. CRDM is a well established technique for gas sensing applications. Nowadays the CRDM is used in optical cavity characterization, and it is successfully implemented in various optical cavities like lithium niobate microdisk [101], strontium fluoride microresonator [102], etc.

In this thesis, we propose to extend the use of the CRDM to retrieve the Brillouin gain coefficient. Performing Brillouin gain characterization inside the laser cavity takes advantage of the light re-circulation of the Stokes wave inside the resonator. As a matter of fact, it strongly increases the effective length of the Stokes wave in the waveguide; this leads to reduce the SBS threshold compared to common gain characterization techniques. This technique could also identify several nonlinear processes that can occur in such a high-Q resonator, for example, thermal drift and Kerr effect [99]. Moreover, the gain coefficient is obtained without the need for material parameter values. Also, this method can be used for characterization of the fiber resonator in order to measure the Q-factor, resonance width, coupling and losses coefficient.

This section is organized as follow: we describe the cavity parameters in section (2.4.1). Then we explain the two characterization approaches, namely the stationary and the dynamic in section (2.4.2) and section (2.4.3) respectively. In these sections, some of the explanations and graphs are taken from the thesis of Stéphane Trebaol [1]. In section (2.4.4), we describe the Brillouin gain coefficient characterization using CRDM and explain the relation between the Brillouin gain and cavity photon lifetime. In section (2.4.5), we make a detailed description of the experimental bench to retrieve the intracavity Brillouin gain coefficient from the CRDM technique. In section (2.4.6), we
discuss the results obtained using the CRDM method. These results have been published in "OSA Continuum" in December 2019 [103].

## 2.4.1 Cavity parameters description

We will describe the resonators in a very general way with the introduction of different notations and parameters such as the quality factor, coupling ratio, cavity losses or gain and resonance width. To introduce the general properties of optical resonators, we study the particular case of a ring resonator. We are going to get the transfer function of our system via the theory of coupled modes [104]. For this, there are two approaches: a frequency one and a temporal one that we relate to stationary and dynamical regimes.

<figure><img src="image_27.png" /><figcaption>Figure 2.11: Schematic view of the resonator. $S_{in}(t)$: input field, $S_{out}(t)$: output field, $u(t)$: resonant field, $\tau_0$: intrinsic photon lifetime, $\tau_e$: coupling lifetime.</figcaption></figure>

The resonator mode under study is described by its lifetime $\tau$ and its field of amplitude $u(t)$ as shown in figure (2.11). The cavity is coupled to a bus waveguide with a coupler. Resonance frequencies are separated by the free spectral range $\Delta\nu_{FSR} = c/(nL) = 1/\tau_L$, where $\tau_L$ is the round trip duration, n is the effective group index of the resonator mode and L is the round trip length. We consider only one resonance frequency $\omega_0$. The resonator mode amplitude $u$ is excited by using an input field $S_{in}$. The total photon lifetime $\tau/2$ in the resonator is related to the coupling and intrinsic losses of the considered resonance:

$$ \frac{1}{\tau} = \frac{1}{\tau_0} + \frac{1}{\tau_e} \qquad (2.17) $$

where $\tau_0/2$ is the intrinsic photon lifetime and $\tau_e/2$ is the coupling lifetime. Similarly, the total quality factor of the resonator is defined as:

$$ \frac{1}{Q} = \frac{1}{Q_0} + \frac{1}{Q_e} \qquad (2.18) $$
where,

$$Q = \frac{\omega_0 \tau}{2} \qquad (2.19a)$$

$$Q_0 = \frac{\omega_0 \tau_0}{2} \qquad (2.19b)$$

$$Q_e = \frac{\omega_0 \tau_e}{2} \qquad (2.19c)$$

$Q_0$ and $Q_e$ are the intrinsic and extrinsic quality factors. $\omega_0$ is the resonance frequency of the cavity. Later, we will show that $\tau_0$ and $\tau_e$ are linked to the cavity losses and coupling ratio, which allowed us to estimate the Brillouin gain coefficient.

According to the damped harmonic oscillator model as described in [1], the excitation signal is $S_{in}(t)$ and the temporal evolution of the mode amplitude in the cavity is $u(t)$.

$$\frac{du}{dt} = \left(i\omega_0 - \frac{1}{\tau}\right)u(t) + \sqrt{\frac{2}{\tau_e}}S_{in}(t), \qquad (2.20)$$

The output signal is expressed as:

$$S_{out}(t) = -S_{in}(t) + \sqrt{\frac{2}{\tau_e}}u(t) \qquad (2.21)$$

If we scan slowly the emitting frequency of the probe signal across the resonator mode resonance (that means in a time longer than the lifetime of the resonator), we obtain the stationary response of the resonator. But if we scan faster (that means in a time shorter than the lifetime of the resonance), then we obtain the transient (dynamic) response of the resonator. Depending on the frequency scanning dynamics of the probe laser line, the shape of the transmitted signal through the bus waveguide will be impacted. We detailed, in the following, two specific configurations of the probe signal and the related information that can be explored from the resonator.

### 2.4.2 Stationary approach:

We assume a monochromatic input signal of the form:

$$S_{in}(t) = S_0 \exp(i\phi(t)) \qquad (2.22)$$

Where $\phi(t)$ is a real function which is equal to $\omega t$, where:

$$\omega = \omega_0 + 2\pi\delta \qquad (2.23)$$
$\delta$ is the detuning from the resonances frequency. Then Eq. (2.20) is solved analytically and we obtained the expression of the transfer function $H(\delta)$:

$$H(\delta) = \frac{S_{out}}{S_{in}} = \frac{1/\tau_e - 1/\tau_0 - i(2\pi\delta)}{1/\tau_e + 1/\tau_0 + i(2\pi\delta)} \quad (2.24)$$

and therefore the transmission $T(\delta) = |H(\delta)|^2$

$$T(\delta) = \frac{(1/\tau_e - 1/\tau_0)^2 + (2\pi\delta)^2}{(1/\tau_e + 1/\tau_0)^2 + (2\pi\delta)^2} \quad (2.25)$$

From this equation, we can plot the transmission spectrum with respect to the photon lifetime, which gives the different operation regimes of a resonator. Figure (2.12 a) shows the transfer function $T(\delta)$ for different values of the $\tau_0/\tau_e$ ratio at resonance frequency $\omega_0$. The curves have a Lorentzian shape with full width at half-maximum of $2\delta_{1/2}$ at the exception of the case $1/\tau_0 = 0$, which corresponds to the transparent regime. This regime is characterized by constant transmission over the spectrum ($T(\delta) = 1$). The different coupling regimes are discussed in the next section.

<figure><img src="image_28.png" /><figcaption>Figure 2.12: (a) Transmission spectrum in steady state for different relative values of $\tau_0$ and $\tau_e$ [1]. (b) Transmission at resonance $T(\delta=0)$ in decibels as a function of $1/\tau_0$ for the different coupling regimes.</figcaption></figure>

### 2.4.2.1 Transmission and coupling regime

Considering the intensity of transmission at resonance $T(\delta = 0)$ in Eq. (2.25 b) we get:

$$T(\delta = 0) = \left( \frac{\tau_e - \tau_0}{\tau_e + \tau_0} \right)^2 \quad (2.26)$$

Figure (2.12 b) shows transmission at resonances $T(\delta = 0)$ for different intrinsic photon lifetimes. The relative values of $\tau_0$ and $\tau_e$ determines the resonator coupling regimes.
The optical transmission spectrum of the figure (2.12a) shows the shape of the cavity resonance.

**Under coupling:** $\tau_0 < \tau_e$, the losses dominate (the resonator transmission is such that: $0 < T(0) < 1$).

**Critical coupling:** $\tau_0 = \tau_e$, the losses are equal to the coupling rate (the transmission drops to zero: $T(0) = 0$).

**Over coupling:** $\tau_0 > \tau_e$, the intrinsic losses are lower than the extrinsic losses. (the transmission resonance is such that $0 < T(0) < 1$).

**Transparency:** $1/\tau_0 \to 0$, the resonator is lossless and coupled system is fully transparent ($T(0) = 1$).

**Selective amplification:** $\tau_0 < -\tau_e < 0$, there is a gain inside the resonator and system acts like an amplifier, $\tau_0 < 0$ but it does not reach the lasing because the coupler leaks are not compensated $|\tau_0| < \tau_e$ (transmission is $T(0) > 1$).

**Laser:** $\tau_0 \to -\tau_e$, the internal gain is such that the losses of the coupler are compensated: $|\tau_0| = \tau_e$. According to Eq. (2.25), the transmission goes to infinity and reaches laser oscillations. Obviously, infinity is not a common physical quantity. A generalized transfer function must be introduced to describe the laser regime [1].

### 2.4.2.2 Q-factor and coupling coefficients

The cavity Q-factor is also expressed in terms of resonance frequency and resonances width:

$$Q = \frac{\nu_0}{2\delta_{1/2}} \qquad (2.27)$$

where $\nu_0 = \omega_0/(2\pi)$ is the optical frequency. If we link this Q-factor from the damped harmonic approach to the one obtained by the matrix approach of Yariv [105] through a high finesse approximation, we obtain:

$$Q = \frac{\omega_0 \tau_L \sqrt{a\rho}}{2(1 - \sqrt{a\rho})} = \frac{\omega_0 \tau}{2} \qquad (2.28)$$

Where, a is the loss term of the cavity (internal cavity loss) and $\rho$ is the field transmission coefficient of the coupler. It is connected to the coupling coefficient ($\kappa$) in the case of a lossless coupler, by $\rho^2 + \kappa^2 = 1$. We may then define an intrinsic quality factor $Q_0$ limited by the losses of the resonator ($\rho = 1$):

$$Q_0 = \frac{\omega_0 \tau_L \sqrt{a}}{2(1 - \sqrt{a})} = \frac{\omega_0 \tau_0}{2} \qquad (2.29)$$
Extrinsic quality factor is limited by the coupling with the access line ($a = 1$):

$$Q_e = \frac{\omega_0 \tau_L \sqrt{\rho}}{2(1 - \sqrt{\rho})} = \frac{\omega_0 \tau_e}{2} \qquad (2.30)$$

Then $\tau_0$ and $\tau_e$ are written:

$$\tau_e = \frac{\tau_L \sqrt{\rho}}{2(1 - \sqrt{\rho})} \qquad (2.31a)$$

$$\tau_0 = \frac{\tau_L \sqrt{a}}{2(1 - \sqrt{a})} \qquad (2.31b)$$

After simplification, we obtain the intensity of round trip attenuation coefficient $|a|^2$ and intensity coupling coefficient $|\kappa|^2$

$$|a|^2 = 1 - \frac{2\tau_L}{\tau_0} \qquad (2.32a)$$

$$|\kappa|^2 = \frac{2\tau_L}{\tau_e} \qquad (2.32b)$$

In Eq. (2.32a), $\tau_0 < 0$ corresponds to an optical attenuation, whereas $\tau_0 > 0$ corresponds to an optical gain. This correspondence will give us the information on Brillouin gain and different coupling regimes.

An important factor, in stationary approach, is the ability to discriminate the coupling regime. In particular, it might happen that equivalent spectral signatures are obtained for over and under coupling regime configurations. In stationary regime over and under coupling regimes cannot be discriminated by a spectral analysis because $\tau_0$ and $\tau_e$ are playing the same role in transmission (see Eq. (2.25)). This discrimination is requested to get the intracavity Brillouin gain, the dynamical approach can overcome this difficulty.

### 2.4.3 Dynamical approach

In case of dynamic approach, we introduced in Eq. (2.22) an angular frequency $\omega$, which is varying in time according to the equation:

$$\omega(t) = \omega_i + \frac{V_s}{2} \times t \qquad (2.33)$$

where $V_s$, the angular frequency sweeping speed. For experimental convenience, we also define $\tilde{V} = V_s/2\pi$, the frequency scanning speed. Considering this change and solving the Eq. (2.20), we can write the mode amplitude as:

$$u(t) = \sqrt{\frac{2}{\tau_e}} S_0 \exp \left( i\omega_0 t - \frac{t}{\tau} \right) \left[ f(t) - f(0) + \frac{1}{i(\omega_i - \omega_0) + 1/\tau} \right] \qquad (2.34)$$
$$f(t) - f(0) = \int_{0}^{t} \exp \left[ i\phi(t') + (1/\tau - i\omega_{0})t' \right] dt' \quad (2.35)$$

Which can be expressed using the complex error a function erf(z) with z ∈ C:

$$f(t) = -\sqrt{\frac{i\pi}{2V_s}} \exp\left[\frac{-i(2\pi\delta_i - i/\tau)^2}{2V_s}\right] \text{erf}\left(\frac{i/\tau - 2\pi\delta_i - V_s t}{\sqrt{2iV_s}}\right) \quad (2.36)$$

Analytical expression of transmission can be expressed as a function of time:

$$T(t) = \left| \frac{2}{\tau_e} S_0 \exp \left( i\omega_0 t - \frac{t}{\tau} \right) \left[ f(t) - f(0) + \frac{1}{i(2\pi\delta_i) + 1/\tau} \right] - 1 \right|^2 \quad (2.37)$$

where we define $2\pi\delta_i = \omega_i - \omega_0$ as the initial detuning. Here, we can see that $\tau_0$ and $\tau_e$ are not interchangeable in the analytical expression of the transmission in Eq. (2.37). For a given coupling regime, $\tau_0$ and $\tau_e$ are fixed, and transmission depends only on the speed of scanning ($V_s$).

Figure (2.13) shows calculated transmission for different scanning speeds in case of a critical coupling configuration. Scanning speed $\tilde{V}_S$ is proportional to the normalized sweeping speed $\tilde{V}_0 = 2/(\pi\tau^2)$, considering a resonance width $(1/(\pi\tau))$ swept in a duration equal to the photon lifetime $\tau/2$. Figure (2.13a) is obtained by slowly scanning the laser frequency of the probe signal ($\tilde{V}_S=0.0075 \tilde{V}_0$). The plotted transmission (Eq. 2.37)) is very similar to the stationary regime. When scanning speed increases, we can observe the appearance of oscillations on the transmission (Figure 2.13b). This ringing is a signature of the transient response of the resonator. Increasing the sweeping speed (Figure 2.13 c and d), enlarge the number of oscillations that can be observed. These oscillations result from the interference between the light coupled inside the resonator at frequency $\nu_0$ and the swept probe signal frequency $\nu(t)$. These oscillations refer to the phenomenon of ring down [106, 107, 108]. This ringing effects enable the phase of the field inside the resonator to be obtained and then the coupling regime to be determined.

Figure (2.14a) shows the transmission of a resonator in the over and under coupling regimes in the stationary or very slow frequency sweeping regime (see Eq.(2.25)). Since $\tau_0$ and $\tau_e$ play the same role in the transmission function T, the two responses are precisely the same. By increasing the sweeping frequency rate, the responses of each coupling regime are sharply different, which allows us to determine the coupling regime as shown in figure (2.14b).

In the following section, the experimental results will be compared to the transient response of the resonator by using the analytical model (Eq.(2.37)) presented above. The
<figure><img src="image_29.png" /><figcaption>Figure 2.13: Transmission of a resonator for different scanning speeds. (a) $\tilde{V}_S=0.0075 \tilde{V}_0$, (b) $\tilde{V}_S=0.3 \tilde{V}_0$, (c) $\tilde{V}_S=3 \tilde{V}_0$, (d) $\tilde{V}_S=30 \tilde{V}_0$ with a $\tilde{V}_0 = 2/(\pi\tau^2)$ [1].</figcaption></figure>

fitting procedure will allow us to deduce the experimental parameters: $V_s$, $\tau_0$, and $\tau_e$. The information of intrinsic losses or gain of the cavity is obtained from $\tau_0$. This helps us to find the Brillouin gain inside the cavity.

### 2.4.4 Estimation of Brillouin gain coefficient

The gain G of the ring cavity can be experimentally evaluated with CRDM. It follows that the coefficient $g_B$ can be calculated from Eq. (2.14). Note that the fiber is polarization maintaining ($k_{polar} = 1$). G is the total gain along one round trip inside the cavity and may be written:

$$G = a^2 = \beta e^{-\alpha_f L} e^{g_B P_{cav} L_{eff} / A_{eff}} = a_{op}^2 e^{g_B P_{cav} L_{eff} / A_{eff}} \quad (2.38)$$

where $\beta$ gives the inner local losses, $\alpha_f$ the fiber losses coefficient, $A_{eff}$ the effective fiber mode area and $L_{eff}$ the usual effective interaction length. $a_{op}^2$ is the intensity round trip attenuation of the cold cavity mode without stimulated Brillouin scattering. $P_{cav}$ is the pump intensity inside the cavity.

As can be seen in Eq. (2.38), when the pump intensity $P_{cav}$ is increased, it allows to progressively compensate linear cavity losses. Thereby, intensity round-trip attenuation
<figure><img src="image_30.png" /><figcaption>Figure 2.14: (a) Transmission in the stationary regime as a function of the normalized detuning in the case of $\tau_0 = 3\tau_e$ and $\tau_e = 3\tau_0$. The two curves are exactly superimposed. (b) Transmission as a function of time in the same two cases with $V_s = 2.25V_0$. Note that the two responses are easily distinguishable [1].</figcaption></figure>

$a^2$ and intrinsic photon lifetime $\tau_0$ can be tuned through the pump intensity keeping the coupler lifetime constant.

We can evaluate experimentally using CRDM, $\tau_0$ and then $a^2 = 1 - 2\tau_L/\tau_0$. The experimental estimation of the Brillouin gain coefficient can be obtained from Eq. (2.38):

$$g_B = \frac{A_{\text{eff}}}{P_{\text{cav}}L_{\text{eff}}} \ln \frac{a^2}{a_{\text{op}}^2} \qquad (2.39)$$

<figure><img src="image_31.png" /><figcaption>Figure 2.15: Spectral overview of the CRDM method: Pump laser line (blue), Brillouin gain curve (green), probed cavity mode (red) and probing laser line (yellow). $\nu_B$ corresponds to the Brillouin shift, where $\Delta\nu_{FSR}$ stands for the spectral spacing (Free Spectral Range) between cavity modes.</figcaption></figure>

We consider the Brillouin scattering in the ring cavity based on a non-resonant pumping system. Figure (2.15) shows the schematic of Brillouin gain in the ring cavity. The
blue line corresponds to the pump signal at frequency $\nu_p$ that produces a Stokes gain (green curve) through the SBS process. In the background of the figure, one can see the periodic distribution of the ring cavity modes (gray lines).

Nevertheless, the perimeter of the resonator, that fixed the free spectral range of the cavity ($\Delta\nu_{FSR}$), and the spectral distribution of the gain curve ensure to favor the amplification of one cavity mode (red color line), that we call the Stokes mode. We consider that the Stokes mode at the frequency $\nu_s$ is at the frequency of the maximum of the gain. Using the CRDM technique, a probe signal $\nu(t)$ (yellow color line on figure 2.15) sweeps across the resonance. The probe is swept with a speed of $V_S$, which gives a transient response. As we have seen before, when the pump input power is increased, the gain will also increase as well as the cavity lifetime. Using the Eq.(2.37), we fit this transient response of the cavity that will give the $\tau_e$ and $\tau_0$ value. By knowing this value we can calculate the Brillouin gain coefficient using Eq.(2.39). This method allows us to characterize the cavity and Brillouin gain coefficients inside the cavity. In the next section, we will discuss the experimental setup for gain characterization and its results.

### 2.4.5 Experimental setup for Brillouin gain measurement using CRDM

In this section we detail the experimental setup and measurement method. It is organized in three paragraphs, which described the fiber cavity, the Brillouin laser setup and the CRDM method. The CRDM experimental setup is presented in figure. (2.16).

The fiber cavity is composed of a $L = 20~m$ polarization maintaining silica fiber distributed as $L_1 = 19~m$ from coupler to circulator port 2 and $L_2 = 1~m$ between circulator port 3 and coupler. The effective interaction length has to be precisely determined to optimized the evaluation of $g_B$. For our cavity configuration, $L_{eff} = L_1 + |\rho|^2 \times L_2 = 19.9 \pm 0.6~m$ where the coupler transmission coefficient $|\rho|^2 = (1 - |k^2|)$ is introduced to incorporate the pump intensity attenuation through the coupler. The effective area of the fiber is $A_{eff} = 84.9 \pm 0.2~\mu m^2$. The cavity is coupled to the input/output fiber by a 95/5 coupler.

The pump is a single mode laser (pump) centered at 1550 nm. The pump intensity is amplified through an Erbium Doped Fiber Amplifier (EDFA). The input power is controlled by a variable optical attenuator. 1 % of the pump is extracted through
<figure><img src="image_32.png" /><figcaption>Figure 2.16: a) Experimental setup for Brillouin gain cavity ringdown determination. EDFA: Erbium-doped fiber amplifier, VOA: Variable optical attenuator, OC : Optical coupler, PC: Polarization controller, BOSA: Brillouin optical spectrum analyzer, PM : Powermeter, PD : Photodetector, OS : Oscilloscope.</figcaption></figure>

a coupler for intensity power monitoring (PM1). The remaining part $P_{cav}$ of the laser pump intensity $P_{in}$ is coupled to the cavity by a circulator. Both quantities can be related through, $P_{cav} = \alpha_{12}P_{in}$, where $\alpha_{12} = 0.758$ is the combined losses of the circulator and connectors. The circulator ensures a single pass to the pump signal in the fiber loop preventing re-injection. For sufficient pump intensity, the stimulated Brillouin Stokes signal is generated in the opposite direction of the pump signal. Extracted Stokes signal is analyzed spectrally thanks to a high-resolution Brillouin Optical Spectrum Analyser (BOSA), while a photodetector records the intensity. This setup is a non resonant Brillouin laser with a threshold of $P_{th} = 61.8$ mW. Lasing properties of this non-resonant Brillouin laser is detailed in the next chapter.

Now we describe the CRDM setup. All the measurements are performed for a pump power $P_{in}$ lower than the pump power lasing threshold $P_{th}$ to be within the validity domain of the technique [97]. The probe signal is provided by a tunable laser (Tunics OM). Before entering the cavity, the signal intensity is controlled by a variable optical attenuator (VOA), and its polarization is adjusted through a polarization controller (PC). The probe laser frequency is positioned close to the Stokes mode, thanks to a 10 MHz - resolution spectrum analyzer (BOSA). The frequency of the probe signal is continuously swept across the resonant Stokes signal by a frequency command allowing to adjust the sweeping speed and the scanning frequency range. A photodetector then collects the transmission of the cavity mode that is observed on an oscilloscope triggered by the
frequency command.

### 2.4.6 Results and discussions

CRDM signals are collected for input pump powers varying from 0 to $P_{th}$. Figure (2.17) presents various CRDM signals collected for pump power ranging from 13.84 to 59.46 mW (red color curves). The normalized transmission is plotted as a function of time. On increasing pump power, the amplitude and number of frequency oscillations increase. To get more insight on those specific temporal signatures, data are fitted using the procedure described above and allows to determine $\tau_0$ and $\tau_e$ as a function of $P_{in}$. As expected the $\tau_e$ value is constant for all the measurements (see table (2.1)).

<figure><img src="image_33.png" /><figcaption>Figure 2.17: Fast scanning profiles recorded from the Brillouin cavity at different pumping power ($P_{in}$) with a scanning speed of 2.8 MHz/µs: The intrinsic photon lifetime ($\tau_0$), transmission and Brillouin gain coefficients are deduced from the analytical fits (red) of experimental curve (black). (a) cold cavity, (b) under coupling, (c) near to critical coupling, (d) over coupling, (e) near to transparency, (f) amplification.</figcaption></figure>

For $P_{in} = 0$ mW, the cold cavity parameters are extracted: $\tau_e = 3.80$ µs and $\tau_0 = 1.20$ µs. Using the definition $Q = 2\pi\nu_S\tau/2$, we get an estimation for the loaded quality factor of $5.6 \times 10^8$. When, $P_{in}$ is varied from 13.84 to 50.14mW, $\tau_0$ increases, modifying the coupling regime from under coupling (cold cavity) towards the transparency regime through the critical coupling and over-coupling. For $50.14 < P_{in} < P_{th}$, $\tau_0$ takes a negative value
corresponding to the amplification regime. The variation of $\tau_0$ as a function of the pump power refers to a progressive compensation of the linear losses until the amplification regime.

<table><caption>Table 2.1: Parameters obtained from the fitting of ringing profiles at different pump powers</caption><thead><tr><th>P<sub>in</sub> [mW]</th><th>τ<sub>e</sub> [μs]</th><th>τ<sub>0</sub> [μs]</th><th>Coupling regime</th><th>g<sub>B</sub> [m/W]</th></tr></thead><tbody><tr><td>only probe</td><td>3.80</td><td>1.20</td><td>—</td><td>—</td></tr><tr><td>13.84</td><td>3.20</td><td>1.56</td><td>undercoupling</td><td>1.71 × 10<sup>-11</sup></td></tr><tr><td>17.22</td><td>3.34</td><td>1.67</td><td>undercoupling</td><td>1.67 × 10<sup>-11</sup></td></tr><tr><td>27.30</td><td>3.43</td><td>2.52</td><td>undercoupling</td><td>1.98 × 10<sup>-11</sup></td></tr><tr><td>39.01</td><td>3.94</td><td>3.82</td><td>critical coupling</td><td>1.78 × 10<sup>-11</sup></td></tr><tr><td>41.70</td><td>3.46</td><td>8.99</td><td>overcoupling</td><td>2.06 × 10<sup>-11</sup></td></tr><tr><td>43.87</td><td>3.87</td><td>0.33</td><td>overcoupling</td><td>2.17 × 10<sup>-11</sup></td></tr><tr><td>48.55</td><td>3.84</td><td>735400</td><td>transparency</td><td>2.03 × 10<sup>-11</sup></td></tr><tr><td>50.14</td><td>3.78</td><td>-7483500</td><td>transparency</td><td>1.96 × 10<sup>-11</sup></td></tr><tr><td>52.02</td><td>3.61</td><td>-0.22</td><td>amplification</td><td>1.98 × 10<sup>-11</sup></td></tr><tr><td>55.87</td><td>3.81</td><td>-0.12</td><td>amplification</td><td>1.91 × 10<sup>-11</sup></td></tr><tr><td>58.51</td><td>3.86</td><td>-6.70</td><td>amplification</td><td>1.96 × 10<sup>-11</sup></td></tr><tr><td>59.46</td><td>3.58</td><td>-5.26</td><td>amplification</td><td>2.00 × 10<sup>-11</sup></td></tr></tbody></table>

Figure (2.18) shows the transmission at the resonance of Brillouin cavity in decibels as a function of $1/\tau_0$ for different regimes, red squares are the estimated values from the gain characterization. The pump power changes the cavity gain; this leads to a change in coupling regime from undercoupling to amplification. From those fitted data, we can retrieve $a^2$ from Eq. (2.38) and then the value of the Brillouin gain $g_B$ by use of Eq. (2.39). Table (2.1) shows calculated gain for different coupling regimes.

Figure (2.19) summarizes the extracted gain values as a function of the pump power. The mean value (red dashed line) is equal to $1.94 \times 10^{-11} \text{ m/W}$ with a standard deviation (shaded green region) of $\pm 1.5 \times 10^{-12} \text{ m/W}$. The uncertainty contribution to $g_B$ related to $A_{\text{eff}}$, $L_{\text{eff}}$ and $P_{\text{cav}}$ gives $0.6 \times 10^{-12} \text{ m/W}$ and do not enlarge the experimentally determined standard deviation of $g_B$. We can then conclude that the Brillouin gain standard deviation is related to the experimental extraction of $\tau_0$. In the next section, we will compare the extracted Brillouin gain coefficient from CRDM with self-heterodyne and pump-probe methods.
<figure><img src="image_34.png" /><figcaption>Figure 2.18: Resonant transmission obtained from the fitting of ringing profiles in the Brillouin cavity. This figure is inspired from article [2].</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>Figure 2.19: Brillouin gain coefficient extracted from the CRDM signal for various input pump powers. The mean value is equal to $g_B = 1.94 \times 10^{-11} \pm 1.5 \times 10^{-12} \text{ m/W}$.</figcaption></figure>
### 2.4.7 Comparison with other methods

To evaluate the relevance of the method, we compare our Brillouin gain estimation through the CRDM technique to other references, for which Brillouin gain bandwidth is experimentally estimated by self-heterodyne [93] and pump-probe [80, 79, 109] techniques. These methods are applied to similar single-mode telecom fibers with around 3 % GeO₂. The Brillouin gain bandwidth is experimentally estimated to be $\Delta\nu_B = 27.5 \pm 2$ MHz. Then, using Eq. (2.11) the gain coefficient can be determined. As mentioned before, this determination depends on the material parameters of the used fiber. Fabrication process conditions (doping, temperature, etc.) affect those parameters. For example, a commercial single-mode fiber in the C band region usually has a GeO₂-doping concentration of a few percent, but this value is usually unknown.

<table><caption>Table 2.2: Comparison of $g_B$ values obtained for single mode silica fibers [5] in various works. SH stands for self-heterodyne.</caption><thead><tr><th>Methods</th><th>$g_B$ value<br/>[×10<sup>-11</sup>m/W]</th><th>standard deviation<br/>[±10<sup>-12</sup>m/W]</th></tr></thead><tbody><tr><td>CRDM [our work]</td><td>1.94</td><td>1.5</td></tr><tr><td>SH [silica value]</td><td>2.45</td><td>1.8</td></tr><tr><td>SH [3% GeO<sub>2</sub> value]</td><td>1.92</td><td>1.4</td></tr><tr><td>Pump-Probe [silica value][79]</td><td>2.29</td><td>-</td></tr><tr><td>Pump-Probe [3% GeO<sub>2</sub> value][92]</td><td>1.98</td><td>-</td></tr><tr><td>Pump-Probe [3% GeO<sub>2</sub> value][80]</td><td>2.07</td><td>-</td></tr></tbody></table>

<table><caption>Table 2.3: Material values for pure and 3% GeO₂ doped silica fibers.</caption><thead><tr><th>Parameters</th><th>SiO<sub>2</sub><br/>[79]</th><th>SiO<sub>2</sub> + 3%GeO<sub>2</sub><br/>[110]</th></tr></thead><tbody><tr><td>Refractive index [n]</td><td>1.45</td><td>1.46</td></tr><tr><td>Elasto-optic coefficient [p<sub>12</sub>]</td><td>0.271</td><td>0.236</td></tr><tr><td>Density [ρ(kg.m<sup>-2</sup>)]</td><td>2200</td><td>2244</td></tr></tbody></table>

We report in table (2.3) material values for pure silica and 3%-doped GeO₂ silica fiber from different journals. Using those parameters, Brillouin gain coefficient estimation gives 2.45 and $1.92 \times 10^{-11}$ m/W (see Table 2.2) for silica and 3% GeO₂ doped silica fibers respectively. The standard deviation, related to $\Delta\nu_B$ determination, is bounded between $\pm 1.4 \times 10^{-12}$ m/W $< \sigma < 1.8 \times 10^{-12}$ m/W. The $g_B$ estimation obtained from the CRDM technique is very close to the one obtained by self-heterodyne using the parameters of a commercial monomode fiber 3%-doped with GeO₂. Upon this example, we see that
<table><caption>Table 2.4: Parameters of a 3% GeO₂ doped silica fibers</caption><thead><tr><th>Parameters</th><th>Values</th><th>References [92]</th><th>References [80]</th></tr></thead><tbody><tr><td>Effective area [A<sub>eff</sub>]</td><td>84.90 µm<sup>2</sup></td><td></td><td></td></tr><tr><td>Length [L]</td><td>26 km</td><td></td><td></td></tr><tr><td>Loss coefficient [α<sub>f</sub>]</td><td>0.20 dB/km</td><td></td><td></td></tr><tr><td>Polarization coefficient [K]</td><td>1.50</td><td></td><td></td></tr><tr><td>Brillouin shift [ν<sub>B</sub>] GHz</td><td>10.87</td><td>21.20</td><td>12.84</td></tr><tr><td>Acoustic velocity [V<sub>A</sub>] m/s (Calculated from 2.7)</td><td>5770</td><td>6011</td><td>5804</td></tr><tr><td>Wavelength [λ<sub>p</sub>] nm</td><td>1550</td><td>828</td><td>1320</td></tr><tr><td>Bandwidth [Δ ν<sub>B</sub>] MHz</td><td>27.50 ±2</td><td>90</td><td>35</td></tr><tr><td>Brillouin threshold [P<sub>th</sub>] mW</td><td>4.70 ±.8</td><td></td><td></td></tr></tbody></table>

the material parameters impact the estimation of the Brillouin gain. Comparison to other works on similar silica fibers (see Tab. 2.2) give comparable $g_B$ values extracted using the CRDM. Pump-Probe gain coefficient values are calculated using Eq (2.11) with parameters shown in tables (2.3) and (2.4).

In conclusion, CRDM method is explained and demonstrated on a Brillouin cavity. We show that CRDM can be used to identify the coupling regime of the cavity by analyzing the intrinsic and extrinsic photon lifetimes. It will help to optimize the laser threshold and laser noise. Also, we show that the CRDM technique can be used in Brillouin gain coefficient characterization in a cavity configuration. Moreover, we measure the Brillouin gain coefficient of SMF-28 fiber with different measurement methods and compared to CRDM; the measured values give a good agreement.

## 2.5 Conclusion

In this chapter, we have reviewed the spontaneous and stimulated Brillouin scattering in optical fiber. We have explained the generality about Brillouin gain and threshold in optical fiber. We have also discussed several approaches for measuring the Brillouin gain coefficient. We have measured the Brillouin gain coefficient using the self-heterodyne method and have validated it using Brillouin threshold determination in SMF-28 fiber. We have demonstrated that CRDM can be used to characterize Brillouin gain coefficient in high-Q resonators. On contrary to usual technique this method through a simple and single-pass measurement gives access to the Brillouin gain coefficient of the material constituting the cavity. We have reviewed the CRDM principle and extracted cavity parameters like Q-factor, intrinsic and extrinsic photon lifetimes, which will help us to
optimize the Brillouin laser threshold and noise performances as it will be explained in the next chapters. The proof of concept has been experimentally demonstrated with a fiber ring resonator. CRDM allowed us to determine unambiguously the coupling regime and to estimate the Brillouin gain coefficient of the material composing the resonator without knowledge of material constants. The self-heterodyne and pump-probe techniques are in good agreement with the CRDM. This cavity ring-down method can be applied to any resonator. The determination of Brillouin gain coefficient in microresonators, exotic material fiber rings and whispering gallery mode resonators for which usual methods do not apply could be done with CRDM.
<figure><img src="image_36.png" /></figure>
# Chapter 3

## Stabilized non-resonant Brillouin fiber laser

### 3.1 Introduction

First Brillouin laser has been studied by Hill *et al.* [111] in 1976. Hill's cavity was rather a hybrid cavity. In addition to the fiber used as a gain medium, it consisted of massive optical components such as lenses, mirrors, and signal separators [112]. It was a Fabry-Perot configuration with a mirror at the end of the fiber. In this first SBS laser demonstration, a cascaded process of up to 14 laser lines have been observed. Later with good stabilization, laser threshold power reached hundreds of mW for Brillouin fiber ring laser [113]. In this free space configuration, nearly 70 % of the signal was lost after one round trip. In 1982, Stokes demonstrated an all-fibered Brillouin laser; fibered tunable directional coupler was used to close the cavity; this allowed him to reduce the cavity losses and the Brillouin laser threshold [114].

The first analytical study of a Brillouin laser was reported in [115]. They give a first model description allowing to express the behavior of the Brillouin laser threshold and Stokes output power. A discussion on the impact of the cavity detuning to the laser performances was also done. A pioneering work has been published in 1991 reporting a 2 kHz linewidth Brillouin laser based on a 20 meter fiber ring cavity [25].

Multi-Stokes Brillouin fiber laser (MBL) have been studied in terms of threshold conditions, efficiency, output and circulating pump power in 1993 [116]. The same group investigated the effects of temperature [117], frequency pulling [118] and frequency sta-
bility [119] of the single-mode Brillouin lasers, after which the dynamics of the Brillouin laser became a subject for several studies [120, 121, 122]. In 2000, Debut *et al.* proposed a theoretical model and predicted that the Brillouin laser linewidth can be narrowed respect to the pump linewidth by the combined influence of acoustic damping rates and cavity losses [123]; they also experimentally demonstrated 12 meters BFL with a laser linewidth, reduced by a factor of 10 [124].

Later, the intensity noise of a BFL was reported theoretically; it was shown that the pumping rate and cavity reinjection are the major parameters of the transfer function of the pump intensity noise for BFL with low finesse cavities [125]. They also predicted that the RIN of the BFL could be lower than that of pump laser RIN for some frequency regions. However, this prediction was not confirmed by their experiments due to mechanical instabilities of the BFL. In 2006, Geng *et al.* proposed an actively stabilized BFL using a PDH frequency locking technique and reported a 20 dB frequency noise reduction [126], later they reported RIN reduction of BFL by experimentally showing that the pump to Stokes noise transfer function acts as an efficient low-pass filter [127, 128].

The development of low-loss optical couplers, circulators, and gratings leads to the development of new Brillouin fiber laser configurations like non-resonant pumping [26], unbalanced Mach-Zehnder interferometer [28], Bragg grating [27], self-injection locking [29] based BFLs. Similarly, hybrid Brillouin/erbium fiber laser has been reported [30]; erbium-doped fiber is inserted inside the cavity, which enables to amplify the Stokes signal thanks to a pump; this configuration reduces the threshold and laser noises, which are reported in different articles [129, 130, 131, 132]. In past decades, Brillouin fiber lasers have been demonstrated in different geometries and materials fibers, like silica photonic crystal fiber [133], tellurite fiber [24, 134, 135, 136], chalcogenide fibers including step-index [137], suspended core [138] and microstructured [139]. Even though some of the first experimental observations of Brillouin scattering were done in liquids [140] and gases [141], after a few decade, first liquid Brillouin laser was demonstrated in integrated liquid-core fiber filled with $CS_2$ [142]. Similarly, $CO_2$ filled Hollow-core fiber-based gas Brillouin laser was reported [143] with 33.2 mW of threshold. These two works have opened a new class of Brillouin lasers.

In 2009, Tomes and Carmon observed SBS in spherical silica resonator where the optical and acoustical modes are whispering-gallery modes (WGM) [144]. Luckily, the higher-order optical WGM-modes have a frequency separation of 11 GHz that corresponds to
the SBS shift at the 1550 nm pump wavelength. Brillouin lasing threshold of 26 µW with 90 % of slope efficiency was observed. In the same year, Grudinin et al. demonstrated the Brillouin laser in 5 mm diameter of CaF₂ WGM resonator, which gives a longitudinal frequency spacing closed to the SBS shift of 17.7 GHz at 1064 nm pump wavelength. The laser threshold is obtained at 3 µW, with significant cascading effects observed at higher pump powers [145]. These WGM Brillouin lasers are obtained by the phase-matching of random spacing between the different optical mode families. In 2012, the group of Vahala introduces Brillouin lasers using a chemically etched ultrahigh-Q wedge resonator on a silicon chip [146]. This resonator is made through a lithographic process, which enables to create microresonator with sub-MHz precision in resonator spacing and high Q-factor (875 millions). Using this resonator, the Brillouin lasing is observed at 50 µW. Also, they obtained a significant linewidth narrowing and cascading effect in this Brillouin laser at 1550 nm pump wavelength [147]. Moreover, these cascaded Brillouin laser signals are used to demonstrate the microwave generation, in which noise performances are comparable to commercial signal generators [148]. More recently low-noise cascaded Brillouin laser was used for the realization of optical gyroscope [149, 150]. In 2014, the group of Vahala demonstrated a Brillouin laser at 1064 nm pump wavelength from an ultra-high-Q silica wedge disk resonator on silicon with a 0.1 Hz²/Hz fundamental frequency noise [151]. Later, variety of microresonator based Brillouin lasers were demonstrated and studied, such as silica-based microsphere [152, 153], micro-rods [154, 155], micro-bottle [156], microbubbles [157] and toroid microcavity [31]. Moreover, microresonator Brillouin lasers were demonstrated with different materials, other than silica as microdisks of barium fluoride [158], strontium fluoride [159], tellurite microspheres [160] and silicon-oil droplet microsphere [161]. However, these microresonators can achieve SBS lasing at low optical power but they are not fully compatible with large-scale optical integration. One alternative is to use an integrated planar ring resonator coupled to a waveguides.

In 2013, Kabakova et al. demonstrated efficient Brillouin lasing using a 20 meters As₂S₃ chalcogenide chip [162]. Due to high insertion loss (5.5 dB) of the chalcogenide chip, a high laser threshold of 330 mW was obtained with a 30 % slope efficiency. Even with very high roundtrip losses, they observed 10 dB linewidth reduction compared to that of the pump laser. In 2018, Gundavarapu et al. reported sub-hertz fundamental linewidth integrated Brillouin laser in Nature Photonics [163]. The laser ring resonator composed of silicon nitride (Si₃N₄) on silica (SiO₂). Although the SBS gain is low in
$Si_3N_4$ [22], the low optical propagation losses of the materials (0.4 dB/m) and 7.43 cm long cavity help to obtain a Q-factor of 28 millions, which leads to a lasing at a relatively low pump threshold of 14.6 mW. This Brillouin laser highlights a 0.7 Hz fundamental linewidth. Even though these microresonators and integrated Brillouin lasers shows a Hz level fundamental linewidth, their integrated linewidth is of the order of pump laser linewidth because these cavities are more sensitive to thermo-refractive fluctuations [164].

This problem can be reduced by a larger mode volume cavity with a short fiber ring res-
onator. In 2019, Loh *et al*. addressed this issue and demonstrated an integrated linewidth
of 20 Hz by using a high-Q resonator consisting of 2 meter polarization-maintaining (PM)
fiber [165]. The group of Loh showed that high-Q fiber resonator SBS laser can be used
as reference sources for portable optical atomic clocks [166].

All the above mentioned Brillouin laser can be classified into two categories: **non-resonant and resonant pumping**. In the first case, the pump is not resonant or it makes a single pass within the cavity, which will be explained in details in this chapter. For the second case, the pump is seeded in a resonant mode in order to circulate inside the cavity, which will reduce the laser threshold when compared to the non-resonant case. The resonant pumping will be studied in the chapter 4.

In the Institut FOTON, resonant and non-resonant Brillouin lasers are studied. In the thesis of Kenny Hey Tow [33], non-resonant Brillouin lasing was realized with suspended core [167] and microstructured chalcogenide fibers [138, 139]. In the first case, non-resonant Brillouin laser was constructed using 3 meters of suspended core $As_{38}Se_{62}$ chalcogenide fiber with silica components. A Koheras laser was used as a pump at 1550 nm wavelength. A lasing threshold of 22 mW with a 26 % slope efficiency was observed. 5 and 6 dB reduction in relative intensity and frequency noise were observed when compared to the pump laser noise respectively [138]. In the second case, 3 meter microstructured GeAsSe fiber was used as a gain medium. At a higher pumping power, two Stokes orders were observed due to Fresnel reflections at chalcogenide fiber facets. The first Stokes in this experiment acts as a pump for the Stokes 2. The linewidth of the pump, Stokes 1 and Stokes 2 were measured to be 4 MHz, 270 kHz, and 20 kHz respectively and approximately 23 dB linewidth narrowing was observed in the cascading process [139].

In the thesis of Schadrac Fresnel [34], were studied both configurations of the Brillouin
laser with silica fiber. In the first case, the non-resonant Brillouin laser was constructed
using a 20 meters silica fiber and lasing was observed at 60 mW of pump power. The RIN of the BFL was studied as a function of pump power. Nearly a 40 dB frequency noise reduction was observed in this cavity. In the second cases, resonant Brillouin laser was demonstrated by 20 meter fiber ring cavity; the pump and Stokes were resonant. Multiple Stokes order were observed and studied and their power dynamics gave similar results as in [116, 168]. The RIN behavior of the first three Stokes orders were studied and it was observed a 20 dB RIN reduction in the clamping regime [34]. Unfortunately the measurement of the frequency noise of the resonant Brillouin laser was limited by the noise floor of the measurement bench eventhough the Brillouin laser was studied in different configurations.

In this thesis, objectives were to identify key parameters in Brillouin fiber lasers and especially their influence on noise properties by taking benefits of the initial work that was done in the lab.

In this chapter, we will focus on the non-resonant Brillouin laser. Firstly, we introduce non-resonant Brillouin laser architecture in section (3.3). After that, the characterization of non-resonant Brillouin laser is explained. In section (3.4) we will describe the experimental setup and will address the issue of mode-hopping in non-resonant Brillouin laser. We propose a new mode-hopping suppression method based on PDH locking. Finally in section (3.6), we will study the frequency noise reduction of non-resonant BFL experimentally and analytically. Moreover, we have realized a mHz linewidth Brillouin laser using cascading of two non-resonant BFL.

## 3.2 Condition for Brillouin lasing effect

Generally, a laser is obtained by inserting an amplifying medium inside an optical cavity. The optical amplification is performed by a population inversion, which favors the stimulated emission. In the case of a Brillouin laser, the resonant optical mode amplification is carried out by stimulated Brillouin scattering. Then the generated Stokes wave is circulating in the opposite direction to that of the pump wave. The laser can oscillate only on cavity modes covered by the Brillouin gain band. Recall that the free spectral range (FSR), which fixes the difference between two neighboring modes, is given by $\Delta\nu_{FSR} = \frac{c}{nL}$; L is the cavity length and n is the refractive index of the medium. To have a resonant mode at the Brillouin-gain maximum, it is necessary that the Brillouin shift
is a multiple of the FSR ($\nu_B = N \times \Delta\nu_{FSR}$, N is the number of cavity modes separating the pump wavelength and the gain-maximum wavelength).

If the number of modes under the gain curve is larger than 1, then the regime is called multi-mode (figure (3.1a)). Generally, it is observed in "long" fiber ring cavities because the FSR is small when compared to the SBS gain bandwidth, multi-mode regime could leads to an unstable laser operation [118, 169]. In a single-mode regime, only one mode will exist under the SBS gain curve (figure (3.1b)), which can be obtained for "short" cavities. The number of resonant modes under the gain bandwidth could be estimated for a side-mode suppression ratio of 20 dB (1 %) to $N_{mode} =2.58 \times (\Delta\nu_B/\Delta\nu_{FSR})$, (see the Appendix C). It turns out that, as the gain is not homogeneous, considering only $\Delta\nu_B/\Delta\nu_{FSR} < 1$ may be enough to have a single-mode operation. It is why we will take this ratio as a reference.

For a stable emission, it is necessary to ensure that there is only one cavity mode under the Brillouin gain curve. In practice, even if this former condition is fulfilled, a resonant mode will shift due to temperature and mechanical fluctuations [117, 119], which could lead to mode-hopping. We can avoid this drawback by applying an active locking to fix the Stokes frequency under the gain curve.

<figure><img src="image_37.png" /><figcaption>Figure 3.1: Distribution of cavity modes under the Brillouin gain curve.</figcaption></figure>

In conclusion, in order to get a single-mode Brillouin laser with an optimized gain, we need to fulfill two necessary conditions. The first one is that $\nu_B$ should be equal to $N \times \Delta\nu_{FSR}$. The second is that the $\Delta\nu_B/\Delta\nu_{FSR}$ should be lower than 1. It is worth mentioning that the Brillouin frequency is defined by the phase matching condition, which depends on the optical wavelength as well as the material properties.
As a matter of fact, temperature fluctuations will affect the generated Stokes frequency by two main mechanisms. The first one is linked to cavity-length fluctuations (due to temperature), which modifies the position of the resonant frequencies of the cavity comb. The second one is related to the acoustic velocity $V_A$ in the medium, which affects the amplitude of the Brillouin shift. Pump frequency fluctuations could also shift the position of the gain maximum. This maximum is separated from the pump wavelength by the Brillouin shift. Then any pump frequency fluctuations will generate the same amplitude of frequency fluctuations of the gain maximum with respect to a resonant mode. Due to these instabilities, a resonant mode, which is located at the gain maximum, may stop lasing at the benefits of an adjacent mode. This phenomenon is called mode-hopping.

Mode-hopping is a phenomenon that a laser experiences with sudden jumps in optical frequency, which are related to the resonator stability or to mode competition in homogeneous medium. The Brillouin laser is operating in a single-frequency regime inside the resonator for some time, but then it suddenly switches to another mode due to some external perturbations. This means that different modes are sharing the pump power for a short period. Mode-hops may be due to temperature variations that will change the central Brillouin frequency of the gain or the cavity resonance frequencies. A "short" fiber cavity will increases the laser threshold; it may even experience this effect if the resonant mode is not well fixed at the maximum Brillouin gain. Moreover, Brillouin scattering and fiber cavity are firmly dependent on temperature. Resonance frequency shifts are related to the refractive index ($n$) and cavity length that are both temperature-dependent. Resonance frequency shifts can be written in terms of thermo-expansion ($\alpha_T = \frac{1}{L} \frac{dL}{dT}$) and thermally induced change in refractive index ($\frac{dn}{dT}$):

$$ \frac{d\nu}{\nu} \approx -\frac{1}{n} \frac{dn}{n} \Delta T - \alpha_T \Delta T \qquad (3.1) $$

With $\alpha_T = 0.51 \times 10^{-6} K^{-1}$ and $\frac{dn}{dT} = 11.6 \times 10^{-6} K^{-1}$ [165]. Thus at a frequency of $193.6 \times 10^{12}$ Hz, the shift is 1.65 GHz for $\Delta T = 1^\circ C$. Brillouin-shift coefficient due to a temperature variation is 1 MHz/K while it is 500 MHz/% for strain coefficient [80].

The Brillouin frequency changes also along with the pump laser wavelength (7 MHz/nm). It is a small change in comparison to the other effects, but this will depend on the pump -laser stability. For a semiconductor DFB laser, the temperature stabilization is around 1/100°C. This implies frequency fluctuations of the order of a few 10's MHz (as typical fluctuations are a few GHz/°C). It is equivalent to a wavelength fluctuations less than the
picometer<fnref n="1" />. It implies a fluctuation for the Brillouin shift of the order of a few kHz. The same conclusion can be drawn for the driving current as the stability is better than 10 µA, which means fluctuations of the order a few ten's of MHz (the wavelength dependence to a change in bias current is a few GHz/mA).

All these effects have influences on mode-hopping in the Brillouin cavity, and it limits the long term single-mode operation. Generally, mode-hopping will be suppressed in resonant Brillouin laser by a servo-loop as it will be explained in the next chapter. Usually, mode-hopping is observed in the non-resonant Brillouin laser because frequency locking scheme were rarely implemented. In this chapter, we proposed a locking method to suppress the mode-hopping in non-resonant Brillouin laser. Before going to the stabilization method, we introduce simple non-resonant Brillouin laser architecture and cavity parameters.

## 3.3 Description of the laser architecture

A schematics of a non-resonant Brillouin laser is shown in figure (3.2). It has been already introduced in the chapter 2 for the purpose of Brillouin gain characterization through CRDM technique. Here, we study the Brillouin lasing of this cavity. The pump laser power is injected through a circulator from port 1 to port 2. The single-mode silica fiber is used as an active medium. It is connected to a coupler to get a part of the laser signal and transmitted pump. One of the coupler ports is connected to the circulator port 3 to close the ring-cavity. It will restrict the pump transmission from port 3 to 2 of the circulator. Then, the pump wave (in blue color) is only performing one round trip inside the cavity. This single-pass pump will induce the stimulated Brillouin scattering process inside the ring cavity. The Stokes wave is generated in the opposite direction. Stokes wave (in red color) is traveling anti-clockwise direction and it will go through the circulator port 2 to 3. It can circulate for several round trips before leaving the cavity.

We consider that the relative position between the cavity modes and the Brillouin gain bandwidth is perfectly overlapping at gain maximum ($g_B(\nu) = g_B$) and also that the cavity is polarization maintained (which implies $k_{polar} = 1$). The losses coefficients of the different components are given in the table (3.1). We consider that the pump and

<footnote n="1">¹A variation of 1 nm is equal to 125 GHz at a wavelength of 1.55 µm</footnote>
<figure><img src="image_38.png" /><figcaption>Figure 3.2: Schematic representation of a BFL cavity with non-resonant pumping showing the different losses coefficients detailed in table (3.1).</figcaption></figure>

<table><caption>Table 3.1: Losses coefficients of each components in the cavity</caption><thead><tr><th>Parameters</th><th>Symbol</th></tr></thead><tbody><tr><td>Losses from port 1 to port 2</td><td>α<sub>12</sub></td></tr><tr><td>Losses from port 2 to port 3</td><td>α<sub>23</sub></td></tr><tr><td>Splicing losses</td><td>α<sub>sp</sub></td></tr><tr><td>Fiber losses</td><td>α<sub>f</sub></td></tr><tr><td>Percentage of the signal extracted from the cavity</td><td>|κ|<sup>2</sup></td></tr><tr><td>Coupler losses</td><td>α<sub>c</sub></td></tr></tbody></table>

Stokes waves will experience total losses, α₁ and α₂ respectively expressed by:

$$ \alpha_1 = \alpha_{12}\alpha_{sp}\alpha_f\alpha_c \qquad (3.2a) $$

$$ \alpha_2 = \alpha_{23}\alpha_{sp}\alpha_f\alpha_c \qquad (3.2b) $$

In the following section, we will characterize round trip losses α₂ by CRDM and describe the Brillouin laser experimental setup.

## 3.4 Experimental set-up

A non-resonant Brillouin laser with 20 m polarization-maintaining (PM) fiber and (95/5) % coupler with 3-port circulator to close the cavity is schematically described in figure (3.3). The cavity is passively stabilized by setting the fiber glowed on a metallic plate. The pump is a tunable C-band source (Tunics OM) associated with an erbium-doped optical fiber amplifier. A variable optical attenuator (VOA) controls the injected power so that the pump has constant noise properties in intensity and frequency as the tunable
<figure><img src="image_39.png" /><figcaption>Figure 3.3: Experimental setup for non-resonant fiber ring cavity Brillouin laser characterization</figcaption></figure>

<table><caption>Table 3.2: Parameters of the non-resonant BFL cavity</caption><thead><tr><th>Parameters</th><th>BFL<sub>20PM</sub></th></tr></thead><tbody><tr><td>L</td><td>20 m</td></tr><tr><td>τ<sub>0</sub></td><td>1.2 μs</td></tr><tr><td>τ<sub>e</sub></td><td>3.8 μs</td></tr><tr><td>α<sub>2</sub></td><td>0.83</td></tr><tr><td>1 - |κ|<sup>2</sup></td><td>95 %</td></tr><tr><td>Q-factor</td><td>5.6 × 10<sup>8</sup></td></tr></tbody></table>

source and the optical amplifier have a constant operating point. A 99/1 % coupler is used to monitor the input pump power. As discussed above, the pump will generate a Brillouin gain and a Stokes wave circulating in the opposite direction to that of the pump. Circulating Stokes power ($P_{S1}$) and transmitted pump ($P_{trans}$) signals are collected on the power-meter placed at the output of the coupler. BFL cavity is characterized by a cavity ring-down method as discussed in chapter 2. Using this method, we extract the intrinsic and extrinsic photon lifetimes of the cavity; this gives the total round trip losses ($\alpha_2$) and Q-factors. The extracted parameters are shown in table (3.2).

### 3.4.1 Characterization of Brillouin laser

In this section, we discuss about the power dynamics of non-resonant Brillouin laser and characterize the Brillouin threshold. The estimation of the Brillouin laser threshold requires the measurement of Stokes power according to the input pump power. Experimental results are shown in figure (3.4b): Stokes output is shown in red and transmitted pump power in black. From this figure, the Brillouin laser threshold is given at 61.8
<figure><img src="image_40.png" /><figcaption>Figure 3.4: (a) Optical spectrum and (b) Power characterization of the 20 m non-resonant BFL cavity</figcaption></figure>

mW. Once the pump reaches the threshold of the Stokes 1, an exponential decrease in transmitted power is observed due to the excess losses from the energy transfer of the pump to the Stokes wave. The Stokes power linearly increases along with the input pump power with 25 % of slope efficiency. It is worth noting that, for the determination of the Brillouin gain coefficient using the CRDM method, the laser is operating below 61.8 mW.

An optical spectrum of the non-resonant Brillouin laser, which is obtained with a high resolution Brillouin optical spectrum analyzer (BOSA), is shown in figure (3.4a). The Stokes component is, offseted by 10.87 GHz from the pump wavelength, which corresponds to the Brillouin frequency shift.

To observe the mode-hopping, we used a high-resolution Fabry-Perot spectrum analyzer because the resolution of the BOSA is limited to 10 MHz, which is in the order of the FSR.

3.4.2 Experimental observation of mode-hopping

Mode-hopping is observed using a high-resolution Fabry-Perot spectrum analyzer with a 2 MHz resolution and 300 MHz FSR. This optical analyzer consists in varying the length separating the two mirrors, over a half-wavelength (half a wavelength corresponds to an FSR), by moving one mirror thanks to a PZT. This change in length of the cavity is so small that it does not affect the FSR of the Fabry-Perot. However, it makes all comb resonant-frequencies shifting along with length variation of the fiber ring cavity.

The detected signal on the photodetector and the triangular PZT scanning amplitude
are displayed on an oscilloscope showing the Airy function. By using the known length separating the two mirrors, the free spectral range is determined and is used to calibrate the horizontal axis of the oscilloscope. Stokes signal is measured and reported in figure

<figure><img src="image_41.png" /><figcaption>Figure 3.5: Mode-hopping of non-resonant Brillouin laser cavity, observed through a Fabry-Perot analyzer in different time intervals (black signal). Red signal shows that we can observe two resonant modes at the same time under the gain bandwidth.</figcaption></figure>

(3.5) using the scanned Fabry-Perot analyser. The figure shows different acquisitions; each time, the resonant mode is suffering a frequency shift. Sometimes two-mode are co-existing under the gain cure simultaneously as shown in red color. In this case, the mode are separated by 10.7 ± 2.0 MHz, which is near to the calculated FSR of the 20 m Brillouin cavity (10 MHz). These observations demonstrate the necessity for an active locking of the resonance frequency by controlling the cavity length or the pump wavelength. The next section describes mode-hopping suppression technique based on cavity-length and pump frequency control.

## 3.5 Stabilized non-resonant Brillouin fiber laser

In this section, we propose a new frequency stabilized mode-hopping suppression method, which improves the frequency noise of the non-resonant Brillouin laser. As seen in section (3.2), the mode-hopping generally observed is due to the shift of the resonant mode, which resonates at the maximum of the gain bandwidth. To suppress this effect we need to stabilize the wavelength of the resonant mode with respect to the gain maximum or the inverse. In 2016, a mode-hopping suppression method on the non-resonant Brillouin cavity was reported [170]. Their approach is based on the phase-locked-loop (PLL), which
freezes the frequency difference between the pump and the Stokes waves. When the length of the Brillouin fiber resonator changes due to thermal and acoustic fluctuations, its resonance frequency drifts relatively to the maximum of the SBS gain. Therefore, the active laser mode, which provides maximum gain, transfers successively from one mode to another.

Measurement of the beating between the Stokes and the pump frequencies includes the detuning between the maximum SBS gain and the resonant frequency associated with the oscillating Stokes wave. Moreover, by freezing the frequency difference between the Stokes and pump, one could automatically stabilize the Stokes oscillating wave with respect to the maximum SBS gain. Effectively, the SBS maximum gain frequency follows the pump frequency. So pump frequency is drifting according to the cavity length drift. It suppresses the mode hopping effect. The locking stability depends on the local oscillator which is used as a reference.

<figure><img src="image_42.png" /><figcaption>Figure 3.6: Experimental setup of BFL cavity with active PDH locking on cavity a resonances. EDFA: Erbium-doped fiber amplifier, VOA: Variable optical attenuator, OC: Optical coupler, PD: Photodetector, PM: Phase modulator, OI: Optical isolator, OF: Optical filter.</figcaption></figure>

We propose a frequency stabilizing method using PDH locking. It is commonly used in resonant Brillouin laser. From our knowledge, it has not been used for non-resonant pumping scheme. The PDH locking method is described in section (1.5.2). In a non-resonant pump scheme, the pump cannot have a resonant mode. We then use a weak reference pump signal called probe, propagating in the same direction than the Stokes. Probe signal is used to lock the pump frequency to a cavity resonance-frequency as shown
in figure (3.6) using the PDH method.

The non-resonant BFL cavity consists of an optical circulator and an optical coupler. The circulator permits a resonant propagation in the backward direction. The laser power is split into two arms, using a coupler. In one arm, the signal is launched into a fiber ring resonator via an optical circulator through an EDFA, and a VOA, in order to generate the Stokes wave. In the signal arm, probe is launched in the same ring cavity but in the backward direction. This probe signal is phase-modulated at 5 MHz to realize a PDH frequency locking mechanism, which maintained the probe in a resonant mode. The benefit of this PDH locking technique is to transfer the stability of the ring cavity to the pump laser. This reduces the drifting (flicker noise) of the pump frequency. Since we can reasonably consider that the Brillouin shift is constant, the generated backward-propagating Stokes in the first arm will be stabilized on a frequency $\nu_{s1}$.

<figure><img src="image_43.png" /><figcaption>Figure 3.7: (a) Beating between pump and Stokes wave with and without mode-hopping stabilization scheme. (b) Optical spectrum of stabilized non-resonant BFL cavity.</figcaption></figure>

For the PDH locking the signal is modulated at a frequency of 5 MHz using a phase modulator. The modulated optical signal is injected into the ring cavity through the coupler. The optical power in the second arm is kept small ($< 1mW$) to have negligible unwanted nonlinear interactions. The output port contains the Stokes signal and the probe signal. A small portion of the output signal is extracted using a 90/10 coupler and is filtered out, using an optical filter to extract the probe signal. The Stokes signal is filtered out from the other port. The PDH probe is demodulated using the PDH driver (from TOPTICA photonics). It generates the error signal that is fed to the control loop that modifies the pump frequency [75, 76]. We can check the disappearance of mode-
hopping using the oscilloscope, as well as the long-term stabilization (more than 1hr). Figure (3.7a), shows the beating signal between pump and Stokes waves in free running (red) with multiple beating due to the mode-hopping phenomena. The black line shows a single frequency beating signal after locking. Moreover, the optical spectrum is measured using the BOSA and is shown in figure (3.7b). For high pump power (202 mW), we observe a Stokes 1 (S1) and Stokes 2 (S2) signals. Signal to noise ratio between Stokes 1 and 2 is 50 dB because the Stokes 2 is not resonant in the cavity. We obtained 0.63 mW output power of Stokes 2 at 300 mW input pump power. The observation of stable Stokes 2 emission is an indirect means to evaluate the stability of the frequency locking scheme.

## 3.6 Frequency noise of non-resonant Brillouin laser

In a conventional laser, light amplification is obtained by population inversion of atoms or molecules. The spontaneous emission associated with the gain medium acts as a quantum noise source responsible for the presence of a minimum spectral width given by the limit of Schawlow-Townes [56]. In semiconductor laser, this limit is a few kHz and can even reach the values of Hertz for gas lasers. This limit is difficult to measure because of the technical noise due to temperature and mechanical vibrations caused by environmental disturbances. The spontaneous emission is not the only source of noise in a conventional laser. The noise of the laser emission also depends on the fluctuations of the pumping system. The intensity noise of a conventional laser is strongly linked to the intensity noise of the pump laser noise. In the case of optically pumped conventional lasers, the frequency noise of the pump does not contribute to the laser frequency noise.

However, in the case of a Brillouin laser, it is not always true because there is a strong coupling between the phase of the emitted Stokes wave and that of the pump wave. Indeed, in a Brillouin process, there is an interaction between three waves via electrostriction. The pump wave interacts with an acoustic wave in order to generate a Stokes wave, (see chapter 2 for details). Then intensity and phase noises of these waves are coupled together.

In the following, we will describe the frequency noise of non-resonant Brillouin laser. We will introduce an analytical model for the frequency noise. Then we compared the analytical and experimental result in different non-resonant Brillouin fiber laser configu-
rations.

### 3.6.1 Analytical frequency noise of non-resonant BFL

Linewidth narrowing of Brillouin laser was experimentally observed by Smith [25] and theoretically studied by Debut [123] within the framework of a three-wave model. It was shown that the phase noise of the Stokes wave is strongly reduced and smoothed under the combined actions of acoustic damping and cavity feedback [124]. The phase noise transfer of the Brillouin laser can be found in [123]:

$$S_0(f) = \frac{-\ln R}{\beta_A - \ln R + i2\pi f} \frac{e^{i\pi f} \sin\pi f}{\pi f} \phi_0(f), \quad (3.3)$$

$$H_1(f) = \frac{-\ln(R)}{\beta_A - \ln(R) + i2\pi f} \quad (3.4a)$$

$$H_2(f) = \frac{e^{i\pi f} \sin\pi f}{\pi f}, \quad (3.4b)$$

$S_0(f)$ and $\phi_0(f)$ are respectively, Stokes and pump phase noises. Eq (3.3) can be factorized into two contributions: $H_1$ and $H_2$. $R$ is the feedback rate of the cavity, which is defined as $R = \alpha_2 \times (1 - |\kappa|^2)$. $\beta_A$ is the acoustic damping rate: $\beta_A = 2\pi\Delta\nu_B/\Delta\nu_{FSR}$. The transfer function of the Stokes phase noise is shown in figure (3.8). The graph is plotted

<figure><img src="image_44.png" /><figcaption>Figure 3.8: Analytical phase noise transfer function for the 20 m fiber length cavity with $R = 0.78$ and $\beta_A = 17.27$.</figcaption></figure>

by taking the value of the fiber length, the Brillouin gain bandwidth and the feedback rate of 20 meters, 27.5 MHz, and 78 % respectively. This transfer function is thus associated with two factors: $H_1$ acts like a low pass filter, which filters out the fluctuations