Noise model for a dual frequency comb beat

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Noise model for a dual frequency comb beat

Thèse

Carlos Andres Perilla Rozo

Doctorat en génie électrique

Philosophiæ doctor (Ph. D.)

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Noise model for a dual frequency comb beat

Thèse

Carlos Andrés Perilla Rozo

Sous la direction de:

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Résumé

Cette thèse porte sur le raffinement d’un modèle du bruit utilisé pour des mesures spectrosco-piques réalisées avec des peignes de fréquences optiques. La majorité des travaux antérieurs utilisaient des peignes de fréquences où le glissement (chirp) est minimisé, en supposant que tout glissement différentiel entre les peignes allait réduire le rapport signal sur bruit. L’hypo-thèse sous-jacente était que l’impact du bruit multiplicatif serait augmenté, le glissement lui permettant d’agir plus longtemps sur le signal d’interférence. Cependant, d’autres recherches indiquaient plutôt contraire : le chirp pourrait améliorer la mesure. Cette thèse cherche à aug-menter la compréhension du comportement du bruit lorsque les peignes ont des glissements différentiels. De plus, celle-ci apporte de nouvelles évidences sur l’utilité du chirp dans ce type de mesure.

À cet effet, nous avons fait une révision bibliographique des modèles du bruit dans les peignes de fréquences optiques. Ensuite, du point de vue théorique, nous avons analysé les effets du chirp sur les bruits additifs et multiplicatifs. Pour le bruit d’intensité, nous avons proposé un modèle phénoménologique décrivant le comportement de l’émission spontanée amplifiée (ASE) dans un laser à verrouillage de mode par rotation non linéaire de polarisation. Les spectres des peignes et leurs battements ont été caractérisés en portant une attention particulière à leur relation avec l’ASE.

La thèse permet de conclure que le chirp différentiel n’affecte pas les niveaux des densités spectrales de bruit. Grâce au glissement différentiel de fréquence, il est possible d’envoyer plus puissance à l’échantillon et ainsi améliorer le rapport signal sur bruit des instruments à peignes de fréquence. D’un autre côté, la caractérisation de l’ASE a établi sa nature non-stationnaire. Elle a aussi expliqué des attributs spectraux qui sont observés régulièrement dans les signaux de battement des peignes.

Finalement, en supposant que l’ASE circule largement dans une cavité opérée sous le seuil, sa caractérisation fournit une méthode pour estimer le déphasage non linéaire que subit le train d’impulsions femtosecondes.

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Abstract

This thesis proposes a noise model refinement for spectroscopic measurements using dual optical frequency combs. Until now most studies centered their efforts on noise characterization using chirp free combs based on an unproved hypothesis: measurements would get worse with chirped combs since multiplicative noises would be present over a longer duration on the interference pattern thus leading to a greater impact. However, at least one experimental result hinted to the contrary: differential chirp would actually improve the signal to noise ratio. This thesis therefore aims at increasing the understanding of noise when a differential chirp is present in a dual comb measurement. The specific goal is to provide new insights about the usefulness of chirp in this kind of measurement.

With this in mind, we conducted a literature review of noise models in optical frequency combs. We subsequently analyzed the chirp’s effect in the presence of both additive and multiplicative noise. The thesis also proposes a phenomenological model to describe the amplified sponta-neous emission - ASE in short pulse lasers mode locked using non linear polarization rotation. Finally the comb spectra and their beat notes are characterized putting special attention to their relation with the ASE components.

As conclusions, we can report that noise power spectral density levels do not change with a differential chirp. Chirping allows sending a greater optical power through the sample, such that the measurement signal to noise ratio can be improved. On the other hand, the ASE characterization established its non-stationary nature and explained very well characteristic features routinely observed in dual comb beat notes that were not fully understood. Finally, assuming the ASE experiences a sub threshold linear cavity allows using theses features to estimate the non linear phase shift experienced by the modelocked pulse train in the laser cavity.

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List of Figures

1.1 A frequency comb centered on the origin. . . 5

1.2 Frequency comb, time domain (above) and frequency domain (below) represen-tations. . . 6

1.3 Carrier and envelope offset – CEO. . . 7

1.4 Simplified diagram of a laser mode-locked fiber ring from Menlo Systems. . . . 8

1.5 A simple setup for dual comb spectrometry. . . 9

1.6 Beat between two modes. . . 10

1.7 Sampled version of an interferogram (IGM) between two frequency combs. . . . 11

1.8 Multibeat diagram for two frequency combs.. . . 13

1.9 Frequency representation of a frequency comb beat.. . . 14

1.10 Two setups for dual comb spectrometry. . . 15

1.11 Representation of the fixed-point model. . . 17

2.1 Generated photocurrent by a pulse pair. . . 23

2.2 Two pulses arrive to the detector. . . 24

2.3 Fiber coupler 2 × 2 schematic.. . . 25

2.4 IQ detection scheme for a dual comb.. . . 26

2.5 IQ detection, a non-ambiguity measurement. . . 27

2.6 Improving dynamic range by chirping. . . 28

2.7 Quantization effects on noise. . . 31

2.8 Additive noise and detection time/bandwidth. . . 33

2.9 Signal to noise ratio as a function of comb power. . . 33

2.10 Shot noise effect. . . 35

2.11 Shot noise signal to noise ratio as a function of comb power. . . 37

2.12 Comb and noise convolution. . . 38

2.13 Shot noise floor versus chirp. . . 39

2.14 A simplified cavity for ASE modeling. . . 41

2.15 Cavity filter magnitude. . . 42

2.16 SNR for the intensity noise as a function of comb power. . . 44

2.17 Beat plus RIN spectra according to detection scheme. . . 45

2.18 Random walk in the pulse pair arrival time. . . 46

2.19 Reference signals and the associated filters. . . 47

2.20 Corrected random walk in the pulse pair arrival time. . . 47

2.21 Jitter to amplitude conversion. . . 48

2.22 Timing jitter noise floor versus chirp. . . 52

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3.1 Setup for a single comb ASE characterization. . . 55

3.2 Lissajous plot for IQ correction for λCW = 1559.9997nm. . . 56

3.3 Raw and corrected spectra for λCW = 1559.9997nm. . . 57

3.4 Phase correction. . . 58

3.5 Fit results for the hump located near to λCW = 1559.9997nm.. . . 59

3.6 Amplitude and width results for 30 measurements. . . 61

3.7 Medium gain (left) and reflexion coefficient (right) for our model. . . 61

3.8 Nonlinear phase shift (left) and optical path difference (right) between the linear and nonlinear in the cavity. . . 63

3.9 Flowchart of the simulator. . . 65

3.10 Single comb PSD. . . 66

3.11 CW–single comb beat. . . 66

3.12 Standard deviation on the detected amplitude for a single comb. . . 67

3.13 CW–comb beat spectrum around λCW= 1554.5nm. . . 67

3.14 ASE pseudo-pulse composition. . . 68

3.15 ASE temporal average. . . 69

3.16 Spectrogram for a dual comb beat. . . 70

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À mes filles Helena, Isabella et Amelia

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Il était une fois. . .

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Remerciements

Malgré toutes les embarras affrontés pendant le développement de cette thèse, j’ai réussi à fermer ce cycle de ma vie. Un peu plus vieux mais avec plus d’expérience académique et personnelle. Plusieurs personnes ont contribué à cette croissance, c’est pourquoi je souhaite les reconnaitre.

Une grande personne, brillante et gentille, qui avec sa capacité d’analyse a guidé le travail académique et avec son grand cœur m’a soutenu aux moments les plus difficiles. Merci à mon directeur de recherche, merci Jérôme.

Je voudrais aussi remercier le groupe de recherche dans lequel j’ai été integré, Jean-Daniel, Simon, Sylvain, Nicolas et Vincent, toujours prêts à collaborer, ils forment une grande équipe. Je voudrais mentionner ma gratitude à Diego Hernández, Ismael Peña et Jesús Quintero, tous de la Faculté de Génie à Bogota qui ont cru que je pourrais y arriver.

Finalement, un merci du cœur à Dayana qui m’a donné le premier coup de fouet pour com-mencer ce long voyage, lequel n’aurait pas été possible sans sa compagnie.

Ce travail a été rendu possible grace au soutient financier de l’Universidad Nacional de Co-lombia et du Gouvernement du Québec, merci infiniment.

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Introduction

With the arrival or refinement of a physical theory and its subsequent application, particu-larly in technological development, the observed deviations or experimental errors feed the improvement of that theory. Frequency combs do not escape this reality. Since their im-plementation, different noise models have been proposed seeking to improve both the combs theoretical understanding and their performance in the laboratory.

Given the nature of their spectra, frequency combs can be used as a spectrographic tool. However, the variations of their primary characteristics (repetition frequency, offset frequency and optical intensity) limit their performance. To overcome these limits, different laboratories followed the scheme outlined in metrological applications: a greater control of the setup and its components. Though, this type of control imposes another limit: it confines the comb to a laboratory. At Université Laval in our research group, another solution was proposed and work was carried out on the subsequent correction of the signals acquired using free running lasers. This way, the combs can be in slightly less controlled environments without sacrificing their performance.

While developing this approach, our group at Laval identified several specificities of the dual comb measurement process. For one aspect, experimental measurements showed that an amount of differential chirp could greatly enhance the signal to noise ratio, potentially by alleviating a dynamic range problem in the acquisition chain [1]. A second aspect is that amplified spontaneous emission produces a very peculiar “hump” in the comb multiheterodyne beatnote. This noise contribution is not consistent with current models. The aim of this thesis is therefore to widen the scope of noise models for optical frequency combs in general and for dual comb experiments in particular.

For this purpose, models for thermal noise, shot noise, intensity noise and timing jitter are analyzed and refinements are proposed. Subsequently, a frequency comb spectrum is measured with emphasis on characterizing the ASE. Based on these measurements, a phenomenological model is suggested to properly explain the observations.

In our opinion, the major contributions of this thesis are two fold. First, it provides a formal explanation of the way differential chirp improves signal to noise ratio in dual comb

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instru-ments. This is mainly achieved in chapter 2 by comparing how signal and noise sources behave in the presence of chirp. Second, the experimental ASE characterization and its associated phenomenological model put into evidence the fact that ASE must in general be non station-ary in mode-locked lasers. This also implies that ASE behaves as a periodically modulated random noise that can be associated to a pseudo comb producing the characteristics “humps” in dual comb beat notes.

The work performed in this thesis contributed to the publication of the following articles and conferences:

• Chemical detection with hyperspectral lidar using dual frequency combs [Sylvain Boudreau, Simon Levasseur, Carlos Perilla, Simon Roy, and Jérôme Genest, Opt. Express 21, 7411-7418 (2013)]. In this work, we measured the optical amplifiers response in large signal conditions leading to establish an empirical limit for the laser probe repetition rate. • Optically referenced double comb interferometry: applications and technological needs [J.

Genest, J. Deschênes, C. A. Perilla, S. Potvin, and S. Boudreau, in CLEO:2011 - Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CThK1.]. In this work, we showed how timing imposed jitter or intensity noise limit in the signal to noise ratio do not change when there is a differential chirp between pulses.

Chapter 1 of this thesis presents the context of the work along with a literature review. It provides a short summary of frequency combs and their use in spectrometry. It also presents the referencing method developed at Université Laval to post-correct dual comb measure-ments. In chapter 2, the noise model is introduced. The literature specific to noise models for frequency combs is also reviewed here. Conventional noise sources, such as shot and thermal noises, amplified spontaneous emission and sampling jitter are considered under the light of interferogram chirp impact. A phenomenological model is also proposed to help understanding the ASE behavior observed and characterized in chapter 3. Experimental measurements are also presented in chapter 3. The main goal here is the characterization of comb ASE in the context of a multiheterodyne beat note. More generally, the noise model of chapter 2 is also validated.

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Chapter 1

Frequency combs: Context

Devised as metrology tools, optical frequency combs have found a place for applications in spectroscopy. Researchers have proposed several methods to increase measurement capabilities. Thus, in this chapter, we want to show the theoretical background of optical frequency combs explaining the mathematical model through their equations and their key relationships. We will start with the basic concepts of frequency combs and their materialization in a mode locked laser. Then, in order to understand how combs are used for spectrometry, we present the beating equations between two combs. Finally, we will explain the corrections we need to perform on the measured signals in order to minimize the effects of disturbances on spectroscopic measurements.

1.1 Frequency combs

For centuries, a number of measurements were made for local trading and the standards were managed by the king, chief or any local authority. As science started to extend its roots and arms, it was necessary to make conventional standards in order to enable both fair trade and scientific experiment repeatability. Standards for length, time, mass and volume were the first ones because of their trading origin; however, over time, other measurements were included according to different needs.

Currently, the Système International d’Unités (SI) comprises all the base units as well as a set of derived ones [2,3]. As our understanding of the world gets deeper, there is a need for better standards and measurement techniques to validate new theories and push applications further [4]. For example, testing fundamental constants as well as theories of relativity or quantum electrodynamics can require measurement accuracies reaching 10−17 _[₅_{]. Time and}

frequency are of particular interest because of our extreme abilities to measure them precisely. As such, more and more units are defined using the time standard and setting a physical constant. For example, the speed of light is no longer defined as a measurable quantity such that length can be directly measured with a clock.

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which is not always easy. Traditionally, in order to measure high frequencies over the GHz range, it was necessary to implement frequency chains that unavoidably propagated errors at each step. Then, if there was a need to determine a visible frequency, e.g. a red color (∼456 THz), it was necessary to create a comparison chain to bridge it down to the frequency of the caesium (Cs) standard (∼9.2 GHz) onto which the SI second is currently based. This chain would have up to ten steps like the one at Physikalisch-Technische Bundesanstalt (PTB) in Germany [6]. The procedure would be the equivalent of trying to measure 50 km using just a one-meter tape, cutting sticks that become longer each time.

In that context, frequency combs were proposed as a tool to directly link optical to electrical frequencies [7]. Frequency combs have brought a revolution in optical frequency metrology, earning part of the Physics Nobel Prize in 2005 [8,9]. Combs have allowed measuring optical frequencies with more than 18 significant digits for wide spectral ranges, e.g. from UV [10] to mid-infrared [11,12]. The self-referenced comb accuracy is currently limited by the quality of the clock used to stabilize the laser repetition rate. Next-generation optical clocks are expected to be superior to existing microwave standards and may become future national standards of time and frequency [5]. When paired with a frequency comb, these clocks provide electrical signals having unparalleled accuracy.

1.1.1 Idealized combs

In the frequency domain, a frequency comb is ideally an infinite arrangement of impulses or teeth. Tooth amplitude and phase are modulated in a particular way and the separation between each tooth is a fixed constant. Mathematically, the spectrum of frequency comb is defined as S(f ) = A(f )1 fr ∞ X n=−∞ δ(f _{− nf}r− f0) (1.1) = A(f )1 fr Xfr (f − f0), (1.2)

where f0 is an offset representing the fact that comb modes do not necessarily fall on an

harmonic number, fr is the separation between adjacent comb teeth, A is a complex quantity

that contains both the phase and amplitude spectral modulation, the term 1

fr is just for

normalization purposes, and Xfr(f ) is a Dirac comb defined like:

Xfr(f ) def = ∞ X n=−∞ δ(f_{− nf}r)

The carrier frequency does not appear explicitly in the equation because it is specified via the modes selected by A(f). From a temporal point of view, the comb is an infinite succession of pulses having a periodic envelope with a period 1/fr.

Fig. 1.1shows a comb with A(f) = exp (−f2_/2σ2

f) with σf as a spectral width. Also in that

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f 0 f 0 f 0 t t t

Figure 1.1: A frequency comb centered on the origin. Frequency domain (left). Pulse formation in the time domain, as more spectral modes are added (right).

gets narrower. Note the pulse is only an envelope without any carrier; this is because the comb spectrum is centered at f = 0. However, for optical frequency combs each tooth corresponds to one longitudinal laser mode in the cavity, and the central frequency νc is about a few

hundreds of THz. Now, if A(f) has the same spectral profile but with a shift to select modes around νc, the temporal pulses will show oscillations around νc modulated by an amplitude

profile having a phase relation fixed by the offset frequency, as shown in Fig.1.2.

Looking for a time domain expression of the comb, we can take the Fourier transform1 _of

Eq. (1.1) and use:

1 fr Xfr (f ) _{⇐⇒ X}F Tr(t) = ∞ X n=−∞ δ(t_{− nT}r) (1.3)

where Tr= 1/fr is the temporal period. The comb time domain expression is then:

s(t) = ˜a(t) ? [ej2πf0t

XTr(t)] (1.4)

where ? denotes convolution, ˜a(t) represents the pulse profile, which, in general, is represented by a complex signal

˜

a(t) = a(t)e−jα(t), (1.5)

1

With Fourier pairs:

g(t) = Z ∞ −∞ G(f )ej2πf tdf G(f ) = Z ∞ −∞ g(t)e−j2πf tdt

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≈ t 1 fr a(t) F · · · · ≈ fr f f0 νc A(f ) |S(f)|

Figure 1.2: Frequency comb, time domain (above) and frequency domain (below) representa-tions. It is important to note that νc does not necessarily fall on a mode.

such that a(t) represents the amplitude, and α(t) the phase dependence. This allows for various pulse amplitude and phase profiles. For instance, this enables a chirp over the pulse duration [13].

Ultrashort pulse stabilized lasers are a practical implementation of frequency combs. Since the theoretical proposal made by Hänsch and his team [5,14] as well as its materialization as a measuring tool made by Udem et al. [15], optical frequency combs have triggered a revolution in optical frequency metrology. In this case, νc falls in about hundreds of THz and fr can

vary from tens of MHz to tens of GHz. The pulse formation mechanism in the laser ensures that the instantaneous frequency of any mode oscillating in the cavity is governed by [7]:

νn= nfr+ f0 (1.6)

where the frequency offset, f0, is a direct consequence of the difference between the phase

(carrier) and group (envelope) velocities in the cavity due its dispersion [13]. Fig.1.3 shows the lag between the carrier and envelope for two successive pulses.

Although the theoretical comb has an infinite spectral width, in practice the number of modes is limited by the gain medium and the residual dispersion in the cavity. A broader spectrum means more workable modes that allow performing measurements in greater spectral ranges. Spectral broadening after the laser cavity output can also be achieved by nonlinearities such as self-phase modulation in crystals and optical fibers including micro-structured and highly nonlinear fibers [7]. Achieving a broad spectrum while managing dispersion implies a

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reduc-≈ t

1

fr φ

Figure 1.3: Carrier and envelope offset – CEO. This offset holds f0 = fr_2πφ [13].

tion in the temporal pulse width which, in turn, enables combs to explore other temporal phenomena such as the study of molecular dynamics [16].

The relation of Eq. (1.6) has been proved to 18 digits for some frequency combs [10]. This means that by measuring frand f0, two electrically accessible frequencies and comparing them

to an atomic clock one can know the absolute frequency of any optical mode.

Different methods for comb generation have been developed. Metrological quality optical frequency combs were first based on mode-locked lasers such as Ti:Sapphire [7,17,18] or doped fiber lasers [19,20,21,22]. The latter option has advantages over bulk solid state sources. One is their low cost and relative simplicity that facilitate the development of turnkey systems. Turning these mode-locked lasers into frequency combs approaching the idealized realization described above does, however, present some challenges. This explains why, during the last fifteen years, the stabilization of both the repetition frequency and the offset frequency has been a very active field of research.

1.2 Fiber frequency combs

In the 60’s doping glass with rare earth trivalent ions made fiber lasers possible [23]. These lasers have severals advantages such as their well-known doping procedures and low loss. Additionally, since they offer a built-in waveguide, they minimize the numbers elements and mechanical alignments. Finally, the intensity dependence of fiber properties is particularly important for certain mode-locking approaches such as nonlinear polarization rotation [21]. In a frequency comb, the deterministic relationship of the spectral modes phase is a crucial aspect. In a mode locked laser this relationship is reached by mechanisms that assist the occurrence and circulation of pulses into the cavity. We can classify these mechanisms as active or passive mode-locking, depending on the existence or absence of external control. Cavities with phase or amplitude modulators belong to the first group, while lasers with a nonlinear element which modulates depending on the intensity belong to the second group. Passive mode-locking is more interesting for frequency combs because it generates shorter pulses and

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WDM EDF λ 2 λ 4 PBS isolator wedge PBS λ 2 λ 4 Pump λ 4 mirror PBS Monitor Exit

Figure 1.4: Simplified diagram of a laser mode-locked fiber ring from Menlo Systems. The gain medium is an erbium doped fiber (EDF). The wedge is to control f0 adjusting the dispersion.

Wave plates control the polarization in the cavity. Nonlinear polarization rotation in fiber helps to maintain mode-locking condition. PBS: Polarizing beam splitter cube. WDM: Wavelength division multiplexer. Blue traces are fiber, red ones are free-space radiation.

hence larger spectra as well as higher peak intensities that allow an easier spectral broadening after the cavity. One approach to mode-lock fiber lasers uses nonlinear polarization rotation (NPR), which in turn relies on the Kerr effect2_{. Concretely, as the pulse propagates along the}

fiber, there is an intensity-dependent rotation of the polarization. Thus, the polarization state of the pulse peak is not the same as the polarization of its wings. We can take advantage of this differential state and filter wings out, keeping only its central part. The lasers we used in this work are fiber ring lasers (model c-comb from Menlo Systems) whose cavity configuration is shown on Fig.1.4. In this scheme the fiber itself provides the nonlinear polarization rotation ensuring mode-locking.

1.3 Beating

The comb spectrum and its constant separation feature also opened doors to comb spec-troscopy, as Yoon et al. [24], van der Weide and Keilmann [25] and Schiller [26] have shown in two different ways. One idea is the use a frequency comb to resonantly enhance the two-photon transition in a particular sample [24]. The other idea is the use of two combs with a small difference between their repetition frequencies. A comb probes the sample and gener-ates a periodic version of its impulse response. The other comb serves as a local oscillator and samples the impulse response in the beating product of heterodyne detection, see Fig.1.5. We

2

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C1 C2 sample D1 t (· · · ) t (· · · ) t (· · · )

Figure 1.5: A simple setup for dual comb spectrometry. Taking the Fourier transform of the measured signal yields information on the sample spectral transmittance. The calibration process is via a measurement without the sample.

will analyze this further in section 1.4, but in order to establish the foundations of the model we will first consider the beat note between only two tones.

If the electric field of each tone is defined by:

s1(t) = a1(t) cos (2πf1t) (1.7)

s2(t) = a2(t) cos (2πf2t), (1.8)

then the detected optical intensity is given by:

i(t) =[s1(t) + s2(t)]2 (1.9) =s2₁(t) + s2₂(t) + 2s1(t)s2(t) (1.10) =a2₁(t) 1 + cos(4πf1t) 2 + a2₂(t) 1 + cos(4πf2t) 2

+a1(t)a2(t){cos [2π(f2− f1)t] + cos [2π(f2+ f1)t]}. (1.11)

As this equation states, the intensity has some terms at twice the optical frequency, but these terms normally are filtered out by the detection system. To illustrate, Fig. 1.6 shows the electric field and the beat note for two modes with a slightly detuned frequency.

Now, we can extend this analysis to a frequency comb. Firstly from a temporal view and later in the frequency domain. In time, when two pulses from different combs (see Eq. (1.4)) arrive at the detector, interference can occur if the pulses are at least partially overlapped. To examine this interference closer, we will analyze a single pair of pulses. Since the arrival period of pulse pairs is usually longer than the detector impulse response duration, an independent detection of each pair is ensured.

Let ˜g(t) be a complex function that contains all temporal information about a pulse, i.e. en-velope and carrier, but for a normalized amplitude such that

Z ∞

−∞

˜

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t t t s1 s2 s1 + s2 |s1 + s2|2

Figure 1.6: Beat between two modes. Red and blue lines represent the electric field of these modes. The purple one is the sum of electric fields. The beat (black line), once detected, has a period given by Eq. (1.10), (f2− f1)−1. The faded gray trace represents the actual intensity

before a slow detection.

This condition of normalization allows expressing the pulse energy as a factor which, in turn, is going to facilitate the noise analysis in next chapter

s1(t) = a1˜g1(t) (1.13)

s2(t) = a2˜g2(t), (1.14)

with a2

i the energy of pulse i. Since these two pulses can arrive at the detector at different

times, we define τ as this difference. In a manner similar to Eq. (1.10), the optical intensity will be:

i(t, τ ) = 1a21|˜g1(t)|2+ 2a22|˜g2(t + τ )|2+ 2√12a1a2<{˜g1(t)˜g2∗(t + τ )}. (1.15)

Here 1and 2 ≈ 1−1are introduced to take into account the effect of combining the combs in

a non ideal beamsplitter or fiber coupler. To detect this intensity, we will use a detector with an impulse response h(t). Thus, the photocurrent will be the convolution between Eq. (1.15) and h(t):

iDEC(t) =R

Z ∞

−∞

h(t0)i(t_{− t}0)dt0 (1.16) where R is the detector responsivity, which is considered constant over the spectral bandwidth of interest. However, since the pulses are short compared to the impulse response, we can consider h(t) as a constant within the integration interval, and therefore, Eq. (1.16) becomes:

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0.5a2 1 0.5a22 0.5(a2 1+ a22) τ1 ∼ 0.5(a2 1+ a22)− a1a2 τ2 ∼ 0.5(a2 1+ a22) + a1a2 τ3

Figure 1.7: Sampled version of an interferogram (IGM) between two frequency combs. The delay between arrival times of two pulses is governed by the difference in theirs repetition rates. For each period, the pulses interfere in a slightly different way than their precedent ones. In the figure, three typical cases are observed. For 1 = 2 = 0.5, from left to right:

without interference; detection is equal to the arithmetic sum of energies, i.e. a half of the pulses initial energy. With destructive interference; detection tends to zero. Finally, with constructive interference; detector senses the initial energy of two pulses. Note that the green pulses represent the detector impulse response which is longer than the optical pulses.

here, we recall that τ is the arrival delay between the pulses and we define the pulse pair energy as w(τ ) = 1a21+ 2a22+ 2√12a1a2< Z ∞ −∞ ˜ g1(t)˜g∗2(t + τ )dt . (1.18)

The last term contains the cross-correlation between ˜g1(t)and ˜g2(t)evaluated at τ. Further,

this term precisely could be interpreted as “the beat note.” According to the delay between the pulses, the interference goes from totally constructive to totally destructive. We will analyze in depth the detection process in section2.2. Meanwhile for illustrative purposes, let’s suppose that s1(t)and s2(t)come from two combs with slightly detuned repetition frequencies,

∆fr = fr1 − fr2. In this way, τ will increment discretely and proportionally to ∆fr for each

new pair. Thus, under the assumption that the temporal profile does not change from pulse to pulse, one measures a sampled version of the cross-correlation in Eq. (1.18), see Fig. 1.7. In the frequency domain, we can extend the single beat note analysis to each comb mode, following a similar interpretation to the one made on the beat frequencies for a pair of teeth [27],

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and using Eq. (1.6). Let

ν1= mfr1+ f01 (1.19)

ν2= nfr2 + f02 (1.20)

be instantaneous frequencies for the nth_{and m}th_{tooth of frequency combs 2 and 1, respectively.}

The detected beat frequency is given by: fRF = ν2− ν1

= nfr2 + f02− mfr1− f01

= nfr2 − nfr1+ nfr1 − mfr1 + f02− f01

= n(fr2− fr1) + (n− m)fr1 + f02− f01

= n∆fr+ ∆f0+ (n− m)fr1. (1.21)

According to this equation, each mode pair produces a detectable beat with the mode of interest falling in the radio frequency (RF) domain, whose frequency depends on the difference of the repetition frequency (∆fr), the difference between CEO frequencies (∆f0), and the mode

index difference (n − m). If we take a look at the last term of Eq. (1.21), it is clear that the beat information is going to repeat twice in every fr1 alias. Fig. 1.8presents this situation.

Finally and for the sake of completeness, another equivalent way of mathematically represent-ing an ideal frequency comb is via a sum of complex exponentials. Each one representrepresent-ing a mode in the frequency domain; where kfr+ f0 defines the frequency of the kth mode, and ak

its complex amplitude. Thus, the electric field can be written in the following form:

s1(t) = ∞ X k=−∞ akexp[j2πt(kfr1 + f01)] (1.22) s2(t) = ∞ X k=−∞ bkexp[j2πt(kfr2+ f02)] (1.23)

for each laser involved in the beating. As si is a real signal, the coefficients for positive k must

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ν (THz)

f (MHz) 0

(n_{− m = 0)} (n_{− m = 1)}fr (n_{− m = 2)}2fr (n_{− m = 3)}3fr

Figure 1.8: Multibeat diagram for two frequency combs. The top axis shows optical frequencies in the range of THz, with four modes for each comb. The bottom axis presents RF, usually in the range of MHz. In the simplest case shown here, modes such that m = n map the information close to f = 0. The next “order” beating, i.e. n − m = 1, maps information closer to f = fr, and so on. Nevertheless, generally, the closest modes have different index,

i.e. m = nmodes do not produce a beating near DC. In the other hand, since the signal is real, the spectrum must have a symmetry and therefore, negative copies, i.e. n − m = −1, −2, . . ., will also appear (dashed arrows).

show that the mixed signal on a photodetector is: iop(t) = 1 2|s1(t) + s2(t)| 2 = ∞ X m=−∞ 1 2(|am| 2₊_|a m|2) +|amb∗m| cos[2πt(m∆fr+ ∆f0) + φa,m− φb,m] (1.24a) + ∞ X p=1 ∞ X m=−∞ am+pa∗m cos(2πtfr1p) + ∞ X m=−∞ bm+pb∗m cos(2πtfr2p) (1.24b) + ∞ X m=−∞

{|am−pb∗m| cos[2πt(pfr1− (m∆fr+ ∆f0))− φa,m−p+ φb,m]} (1.24c)

+

∞

X

m=−∞

{|am+pb∗m| cos[2πt(pfr1+ (m∆fr+ ∆f0)) + φa,m+p− φb,m]}

!

, (1.24d)

such that φa,m and φb,m are the argument of am and bm respectively. The first line, (1.24a),

shows the comb’s average power and a modulation that depends on the value of the differ-ence between the repetition frequencies (∆fr) and difference in the offset frequencies (∆f0).

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p = 1 p = 2

0 fr 2fr

p = 0

Figure 1.9: Frequency representation of a frequency comb beat. The blue line represents the DC value and the first replica (bell-shaped) of the beat (p = 0) according to Eq. (1.24a). The ocher hue lines represent the repeated frequency harmonics (p ≥ 1), Eq. (1.24b). The red and green graphics are the successive beating replicas around the harmonic, Eqs. (1.24c) and (1.24d), respectively.

a Michelson interferometer. The second line, (1.24b), reflects the harmonics of repetition fre-quencies. Finally, the last two lines represent the beating between the combs, but replicated to the left (1.24c) and to the right (1.24d) of each harmonic indexed by p in (1.24b). Note that the term p = 0 is precisely (1.24a). In other words, the signal iop(t) is the sum of beats

from the comb’s teeth, shifted and controlled by p and thus, it is in agreement with Eq. (1.21) where p = n − m. In Fig. 1.9, we can see the frequency representation of such a beat.

Up to this point, we have seen how frequency combs work and how they interfere to produce a sampled version of an interferogram (IGM). In the next section, we will see how to use them to retrieve optical spectral information into an electrical spectrum.

1.4 Dual frequency combs spectrometry

The interaction between light and media can be analyzed through spectroscopy. The concept behind this technique is to filter a light beam with an optical sample and then examine the resulting light spectrum. For frequency combs depending on the setup, one or both combs are filtered by the sample. Because, as seen in Figs. 1.8and 1.9, each optical teeth pair are mapped to electrical frequencies via the beat note, it is possible to retrieve optical spectral information from the electrical spectrum.

Now, we will present two possible setups for measuring spectral features of an unknown sample. In Fig.1.10a), detector D1 measures the beat note between two combs filtered by the sample,

while detector D2 measures the unfiltered version. In this way, the signal from detector D2

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C1 C2 sample D1 D2 a) C1 C2 sample D1 D2 b)

Figure 1.10: Two setups for dual comb spectrometry. The sample is probed by a) two combs or b) only for one. Depending on what spectral feature is going to be estimated, the setup is chosen.

this method can estimate the sample transmittance but no phase information is measured. In Fig. 1.10 b), only one comb probes the sample and the other one serves as a local oscillator. With this approach, only a half of the power is filtered by the sample; this means a sensitivity loss in the transmittance estimation. However, the phase information is accessible [28]. Several research teams have made proofs of concept using different methods, and all have given valid results. One of the first reports [18] showed the use of two mode-locked Ti:Sapphire lasers. In that setup, they generated harmonic combs, i.e. f0= 0, from frequency differences produced

by nonlinearities in GaSe crystals. The lasers central wavelength was around 800 nm and their repetition frequencies were slightly different (∆fr ≈ 2 Hz). For the first comb, nonlinearity

in the crystal produced a new comb with each tooth at the frequency difference between two teeth of original comb, i.e. (nfr + f0)− (mfr+ f0) = (n− m)fr. For some m − n, this

difference falls in mid-infrared and generates an harmonic comb, around 2.4 µm. In a similar fashion, the other comb had frequency teeth at (n − m)(fr− ∆fr). Since both combs were

harmonic, they did not have any problems with f0 control. The spectral resolution limit,

about 13 cm−1_{, was imposed by f}

r stabilization. Subsequent results showed a resolution of

2 cm−1 _[₂₉_{]. Since then, other groups have followed this idea and resolutions increased to}

0.033 cm−1 [30] and 0.003 cm−1 [31], mainly by improving control on fr. All these techniques

have used Ti:Sapphire lasers as sources. The use of mode-locked fiber lasers, but also the control of f0, enabled Coddington et al. [32] to report a resolution of 0.003 cm−1 with a

simpler setup.

These setups tried to control the key parameters of combs, e.g. fr or f0; however, their

implementation, maintenance, and use were neither easy nor cheap, they were actually rather expensive, relegating them to specialized laboratories.

In contrast to this trend and almost simultaneously, J. Genest’s group at Université Laval reported how to generate a reference grid for the spectrum obtained from the beating of two fiber combs that are not highly stabilized [33]. This approach reduces the complexity of the

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installation and increases robustness to fr and f0 fluctuations. The spectral resolution was

limited to 0.06 cm−1 _{by the length of the fiber Bragg gratings used to measure the correction}

parameters [33]. A new correction scheme proposed by Deschênes et al. [1] allowed stepping the resolution down to 0.003 cm−1_{, limited by the comb’s repetition rate.}

Combinations of classic spectroscopic ideas with frequency combs such as the cavity-enhanced technique that can offer high sensitivities and resolutions can also be found in the literature [34,

35,36,37]. The reported resolutions were within the range of 0.01 cm−1 _{and 0.15 cm}−1_{. An}

article by Thorpe and Ye [38] is an excellent reference for this technique.

In comb spectroscopy, the laser linewidth fixes the ultimate spectral resolution. However, the above methods had a spectral sampling limited by the pulse repetition rate. Jacquet et al. proposed a method [39] to overcome this limitation. They acquired several interferograms, changing the combs parameters between each measurement such as to move the modes “po-sitions,” allowing a sort of spectral interleaved sampling. With this method, they obtained a resolution of 7.7 × 10−8 _cm−1 _{(2.3 kHz). Shioda et al. also proposed a similar idea [}₄₀_{]. They}

generated a comb with a variable repetition rate; it could be adjusted using an optical modu-lator, performing a scanning between 3 GHz and 13 GHz. For each step, the comb probes the sample, and its impulse response is detected in a heterodyne way with a frequency-tunable laser. The reported resolution was < 0.00003 cm−1 _{(1 MHz).}

As any measurement process, dual comb spectrometry is not errorless. Different kind of noise sources affect each reading with undesirable fluctuations. Thus the final measurement will be reported with some uncertainty. In the next section, we will review some aspects that contribute to that uncertainty.

1.5 Frequency fluctuations in a comb

One of the first reports on the frequency noise analysis in a mode-locked laser regime presented by Ho [41] showed that the comb modes behaved as replicas of one another spectrally speaking, i.e. all modes share the same spectral width. At first order the comb modes acquire the same amount of phase noise and, although it was developed for a single mode, the Schawlow-Townes equation can be modified to calculate that linewidth [42]. However, according to Eq. (1.6), fluctuations on f0 and fr should have a different impact. For instance, changes of

f0 affect all modes in the same way, but variations on fr lead to a larger impact at high n.

Nevertheless, if one restricts n to a narrow range, e.g. around the carrier, one could say that all modes experience similar fluctuations. Shortly after Ho, von der Linde measured the spectrum experimentally and explained it with a simple but limited model [43] in which they only considered the intensity fluctuations and a temporal jitter over an ideal mode-locked laser. The findings of both Ho and von der Linde measurements were corroborated with a complete theoretical analysis by Hauss and Mecozzi [44] which provided the framework

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Undisturbed f0 FP νc Shifting disturbances f0 FP νc Stretching disturbances f0 FP νc

Figure 1.11: Representation of the fixed-point model. In this figure, parallel blue lines repre-sent comb modes, not to scale, and FP marks the fixed-point location. Modes away from FP experience larger fluctuations due to a stretching of the scale.

for more recent work. In that work, they discriminate noise sources, like pump, mirror-position, and beam axis fluctuations. And they also include effects from self-phase modulation, group velocity dispersion and saturable absorber. The model is developed for soliton mode-locked lasers with a fast saturable absorber. Paschotta’s numerical model [45, 46] and its subsequent analytical calculations [47] have highlighted that Hauss’ work could be extended beyond its initial limitations. Moreover, when he presented a guide for designing a laser mode locking regime, he revealed the limits and interactions between different noises on the comb, i.e. phase, intensity and timing jitter, and settings point, such as the cavity gain. While these works focused on the noise inside the cavity and seeking its limit, the group led by Telle [48] suggested a little more convenient interpretation for those who use the comb as a tool rather than as a design goal. For a particular frequency or phase noise source, its effects on the comb can be fully determined by a frequency shift (associated to noise on f0) and a stretch of the

frequency axis around a given fixed point (linked to frfluctuations). This so-called fixed-point

model is a useful tool to analyze and predict the mode’s behavior. According to Newbury et al. [49], intracavity noises are adequately represented by this model. On the other hand, some noise sources outside the laser cavity affect modes independently, so they do not satisfy this condition; consequently, experimental observations can deviate from the model.

This model’s main idea is to see the comb’s frequency scale as an elastic tape which is shifted and stretched around a fixed point. Shifting variations affect all modes in the same way while stretching around the fixed point does not. Near the fixed point variations are small but gradually as one moves away, they get stronger. According to this model, the fixed point position depends heavily on the noise source and how perturbations are seen. For example, disturbances with a fixed point at low frequencies would have a greater impact on the carrier frequency fluctuations, one example of this would be cavity length fluctuations. Fig. 1.11

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point near the carrier frequency. This explains why, at first order, all optical mode appear to have similar fluctuations. The model fits completely with our understanding of combs. There are only two parameters to know and control, f0 and fr. Different teams have used this

fixed-point model with a high agreement. For example, McFerran et al. [50,51] have based their control strategy on this model for stabilizing the comb. Moreover, Benkler et al. [52] showed experimental results validating the model.

As we have already stated, for frequency combs, f0 and fr determine all the behavior of the

frequency scale. The stability and knowledge of these parameters thus affect the performance. In the next section, we will review a method to reduce the uncertainty due to the lack of a perfect knowledge of f0 and fr.

1.6 Referencing method

In a dual comb experiment, any unaccounted deviation of ∆fr or ∆f0 will introduce an error

in the measurement. Minimizing these deviations is the reason why it seemed necessary to use stabilized lasers, which can be complex and expensive. On the other hand, it is possible to make a post-correction using reference signals that have been recorded simultaneously with the signal of interest, as Giaccari et al. have suggested [33]. This correction is entirely similar to measuring the mirror speed signal in a Michelson interferometer, but it should be noted that this interferometer uses only one signal (speed correction), while for the combs, two signals are necessary (one for fr and one for f0). In this section, we will briefly discuss the referencing

system used in this work, in order to correct the measured signals.

In order to model the nuisance effects, one hypothesis holds that all teeth disturbances can be modeled only with variations of fr and f0 –a hypothesis that has been tested by different

models and measures [41,48]. In this context, we need to track two teeth pairs. The first one gives information about the evolution of ∆f0and the second, which should be spectrally as far

as possible from the first one, helps in measuring the change in ∆fr. The idea of referencing

is to isolate the beat of these teeth pairs and to compare them with ideal beats, i.e. combs with perfect fr and f0 and then, estimate the fluctuations and correct for them.

It can be shown [1] that after optical filtering centered at fc of the combs and analyzing a

single pair of pulses, the electrical signal in complex notation at the output of a photodetector is:

sd(t) =hd(t− Tr1)A(∆Tr) exp(j2πfc∆Tr+ j∆φ) (1.25)

where hd is the detector impulse response, A is the baseband autocorrelation of the filter’s

impulse response, ∆Tr is the difference between pulses arrival times, and ∆φ their phase

dif-ference. The values of ∆Trand ∆φ contain information about the relative comb disturbances.

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if ∆f0 were constant, ∆φ would be expected to grow linearly. However, different sources of

disturbances affect the behavior of combs, causing drifts, jitter and noise on ∆Tr and the

CEO.

Successive pulse pairs arrive at times intervals greater than the detector response time. This means that there will be no interaction between them. Thus, the electrical pulse train can be expressed as the sum of k independent expressions, such as Eq. (1.25). Besides, if sampling is synchronized with the arrival of each pulse of the first comb i.e. Tr1(k), the reference signals

and the wide band beat note, which has an autocorrelation function in discrete time, Am,

becomes:

rd1[k] =hd(0)A1(∆Tr(k)) exp(j2πfc1∆Tr(k) + j∆φ(k)) (1.26)

rd2[k] =hd(0)A2(∆Tr(k)) exp(j2πfc2∆Tr(k) + j∆φ(k)) (1.27)

sm[k] =hd(0)Am(∆Tr(k)) exp(j2πfm∆Tr(k) + j∆φ(k)). (1.28)

Firstly, the amplitudes of reference signals, A1 and A2, must be robust over all the desired

inter-pulse delays to allow the extraction of reliable information. A single electrical beating mode is thus preferably isolated for each reference signal. Also, from the reference equations, it may be noted that the phase of these signals contains the correction information, which means that it is necessary to extract the phase of the measured reference signals.

The first step to correct the signal sm is to eliminate the dependence on ∆φ by subtracting

the extracted phase of rd1 (or of rd2):

smC1[k] =hd(0)Am(∆Tr(k)) exp[j2π(fm− fc1)∆Tr(k)]. (1.29)

Secondly, we need to resample on a grid where the increment of ∆Tr is constant. This grid is

precisely given by the corrected phase (without ∆φ) in the other reference signal:

G[k] = exp[j2π(fc2 − fc1)∆Tr(k)]. (1.30)

Finally, if we want to increase the signal to noise ratio by averaging, the corrected signal must undergo two final adjustments. In one hand, in order to match the peak of the IGM or the zero path difference (ZPD) for all measurements, a time shift is necessary. On the other hand, because of the CEO phase, we also need to correct the phase at ZPD, which may be different in each interferogram.

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Chapter 2

Noise model

Different noise sources influence the beating between two frequency combs. Each comb has its own noise: intensity and phase, and also a timing jitter, among others. Also, measurement systems contribute to a rising in noise levels. Shot noise and detector noise belong to this category. Finally, the acquisition system noise could affect the results through quantization noise. In this chapter, we will study these noises and their impact on beat measurement.

2.1 Noise in dual frequency comb spectroscopy

When modelling noise in a dual comb spectrometer, there are two aspects we need to take into account. The first one deals directly with noise sources at the generation point and within the resonant cavity output. Although we have not worked directly with the design and construction of an ultrashort pulse laser, this is important for us because it helps to understand their origins and possible correlations between the parameters that define the comb (fr and

f0). The second aspect is related to out-of-cavity noise, e.g. effects of the environment or

electronic detection, and how this limits the spectroscopic measurement. As some sources produce similar detectable effects, they are studied here as only one noise; this leads us to classify by their effects rather than by their origin. So in this classification, we find terms such as phase noise, timing jitter, and relative intensity noise – RIN.

A team at NIST published two studies on frequency combs spectroscopy which included a noise analysis [53, 54]. In the first article, Coddington et al. [53] show a detailed description of their dual comb system with emphasis on signal to noise characteristics and aspects that limit the spectral resolution. In the second work, Newbury et al. [54] made a general study of how different sources of noise restrict the sensitivity of a frequency comb spectrometer. This work is mainly focused on additive noise, i.e. detector noise, intensity noise and shot noise, assuming that frand f0are strongly controlled and that there is no differential chirp between the combs.

In that case, the multiplicative noise was assumed negligible compared to the additive noise. The parameter used for the study was the signal to noise ratio (SNR) as a function of comb’s

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power. In that theoretical analysis, they divided the comb bandwidth into Nd spectral bands,

which were coupled to the same number of detectors. The idea behind this division was to reduce excess source noise and dynamic range limitations. Additionally, each band was split into F sub-bands so that they could do parallel detection (F = 1) or sequential detection (F > 1). In the last case, the detection time for each sub-band was reduced by a factor 1/F . As a conclusion, for schemes that reduce the input power to the detector, e.g. Nd > 1 or

F > 1, there was an advantage for high power as a result of the intensity noise distribution. In practice, however, the assembly becomes quickly complex. Thus, according to Newbury et al. the product N_dF should remain unitary. For low power (∼1 µW), the detector (thermal noise) imposes the SNR limit. For the power range between 10 µW and 100 µW, shot noise is the limit. For greater power, the restriction is related to intensity noise or to the detection system dynamic range. Newbury et al. also used and adapted the quality factor coined by Bell [55] as a figure of merit for comparison between comb-based spectrometers. The factor is defined as M/σH with an acquisition time of 1 s, where M = (spectral range)/(resolution)

is the number of spectral elements resolved and σH is the averaged uncertainty across the

spectrum at each spectral element. In their work [53,56], this factor was 2 × 106 _Hz1/2_{. For}

Bernhardt et al. [37], it was 1.7 × 106 _Hz.1/2 _{For the first version at Université Laval [}₃₃_],

it was 0.25 × 106 _Hz1/2_{, and for the second [}₁_{], it was 4 × 10}7 _Hz1/2_{. This last value is}

even beyond the limit given by Newbury et al. for instruments using fiber lasers (107 _Hz1/2_).

This difference leads us to reconsider the validity of their assumptions for a system whose a posteriori correction counterbalances its lack of stabilization, such as that used by Deschênes et al. [1]. Also, in that work they have shown experimentally that some dispersion could help improve the quality of the measured spectra. One possible explanation for the high merit factor resides in the use of dispersion to chirp the pulses such as to reduce the dynamic range problem.

Recently, the NIST team has reported [57] the presence of wake instabilities in a mode-locked laser pulse based on a slow saturable absorber [58,59]. The occurrence of these instabilities is attributed to a net gain window in time created due to the difference in the longer saturable absorber response time and the shorter pulse duration. This window allows the ASE to grow, eventually becoming a new pulse. Similarly, Wang et al. [60] computationally modeled these types of pulses and described the wake mode sidebands that are experimentally observed in the comb output spectrum. Unlike these lasers, the sources used in this thesis use nonlinear polarization rotation, which acts as a fast saturable absorber maintaining the mode locked condition. In this case, the temporal net gain window is not large enough to create significant instabilities. However, the presence of non-stationary noise of increasing variance is experi-mentally observable as we will show in Chapter 3. In section 2.7, we propose an empirical model for these phenomena.

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shot noise and noise in the laser intensity, can also have an impact on the timing and phase noise. To see this effect qualitatively and following the line of Deschênes [61], consider a sinusoidal signal that has an additive noise n(t):

s(t) = a· sin(2πft) + n(t) (2.1)

where a n(t). If we are interested in knowing the true zero crossing, it is necessary to solve for t at s(t) = 0:

0 = a· sin(2πft) + n(t) 0≈ a · (2πft) + n(t)

t =− n(t)

2πaf. (2.2)

It can be seen that variations in additive noise are reflected as variations in time, with a standard deviation proportional to the inverse of the slope of the signal:

σt=

σn

2πaf. (2.3)

From a similar analysis, it is possible to show that the noises associated with the phase and jitter timing can induce additive variations.

2.2 Theoretical signal

In this section, we develop an expression for the generated electrical charge when a pulse pair arrives on the detector. First, we define the electric field of pulses that produces a beat. As seen in Chapter 1, the intensity for a single interfering pulse pair is given by:

i(t) = 1a21|˜g1(t− τ1)|2+ 2a22|˜g2(t− τ2)|2+ 2√12a1a2<{˜g1(t− τ1)˜g2∗(t− τ2)}, (2.4)

and the photocurrent produced by the photodetector may be seen as a convolution between the detector impulse response h(t) and the optical intensity:

iDEC(t) =R

Z ∞

−∞

h(t0)i(t_{− t}0)dt0 (2.5)

These are just Eqs. (1.15) and (1.16) rewritten here for clarity. Now we write ˜g1(t)and ˜g2(t)

as:

g1(t) = ˜a1(t− τ1) exp [j2πνc(t− τ1)] exp (jφ1) (2.6)

g2(t) = ˜a2(t− τ2) exp [j2πνc(t− τ2)] exp (jφ2). (2.7)

where ˜ai are the complex envelopes, νc the optical carrier, φi the initial phases and τi the

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t i(t)

τA

b

q

Figure 2.1: Generated photocurrent by a pulse pair. Assuming a unitary impulse response h(t), the area under the curve is the total electrical charge q given by Eq. (2.8).

Since, as mentioned before, the pulses are much shorter than the detector’s response, one may approximate the optical intensity as a Dirac pulse δ(t) producing a total charge q:

Ri(t) → qδ(t − τA)

where τA= (τ1+ τ2)/2is the mean arrival time. The charge q can be calculated as:

q =R Z ∞

−∞

i(t)dt, (2.8)

Moreover, the detector output photocurrent will be: i(t) = q

Z ∞

−∞

h(t0)δ(t_{− τ}A− t0)dt0

= qh(t− τA). (2.9)

This equation states that the photocurrent produced by a single pulse pair will show as a detector impulse response whose height and area will be modulated by the total charge. This is represented schematically in Fig.2.1where the area of h(t) is assumed unitary for simplicity. The information of the sampled dual comb interferogram, therefore, lies in the total charge in each photodetected pulse pair and hence in the area of each detected impulse response. Substituting Eq. (2.4) in Eq. (2.8), the total charge as function of the arrival times is given by: q(τ1, τ2) =R1a21+ 2a22+ 2√12a1a2< {C(τ1, τ2)} (2.10) where C(τ1, τ2) = Z ∞ −∞ ˜ a1(t− τ1)˜a∗2(t− τ2) exp [j2πνc(τ1− τ2)] exp [j(φ1− φ2)]dt. (2.11)

If we assume that the temporal profile is invariant, i.e. the shape does not change from one pulse to another, we can rewrite this integral as a function of the differential delay ∆τk =

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t

∆τ ∆φ

Figure 2.2: Two pulses arrive to the detector. ∆τ is the temporal difference between envelopes, and ∆φ is phase difference between the CEO of combs.

(τ1− τ2)k and differential phase ∆φk = (φ1− φ2)k, see Fig. 2.2, thus,

Ck(∆τk, ∆φk) = exp (j2πνc∆τk) exp (j∆φk)

Z ∞

−∞

˜

a1(t)˜a∗2(t + ∆τk)dt, (2.12)

where k counts the pulse pairs and is a reminder of the sampled nature of the IGM. Finally, the electrical charge for each arrival is:

qk(∆τk, ∆φk) =R1a21+ 2a22+ 2√12a1a2< {Ck(∆τk, ∆φk)} . (2.13)

In this equation, the two first terms are proportional to the energy of each pulse. This energy corresponds to the light that is not modulated interferometrically and is not normally of interest. It should be noted, however, that these terms can make great contributions to shot noise and intensity noise. This can be significant because those two first unmodulated terms are usually larger than the signal of interest carried by the last term. Eq. (2.13) is the starting point to analyze the variations of the electric charge according to the noise sources. In the following sections, after a brief review of detection schemes, we are going to estimate the charge variance for different kind of noises.

2.3 Detection schemes

Several elements work together to make possible the conversion of light to digital signals that can be processed. At least three elements can be found. A photodetector, an amplifier and an analog to digital converter (ADC). Although there are many kinds of detectors, our detection element is usually a photodiode which converts light into electrical current. Also, it is possible to find either optical or electrical setups to improve the detection process. In this section, we will deal with these elements and setups that will have different impacts on the noise analysis.

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s1 s2 s3 s4 Port 1 Port 2 Port 3 Port 4

Figure 2.3: Fiber coupler 2×2 schematic. Although the coupler is bidirectional, here we define ports 1 and 2 as input and ports 3 and 4 as output.

The simplest detection scheme for an interference signal uses a single optical detector and a DC-coupled amplifier. For this configuration, light is mixed with a beam splitter or coupler and one output port is aimed to the detector. In this case, the small interferometric signal must be measured in the presence of the two larger unmodulated terms. This is often aggravated by the fact that amplifier saturation and nonlinearity set a power limit on the detector. In general this means that the detection chain must have a good dynamic range. The unmodulated terms can also add excess noise.

Balanced detection serves as a way not only to get rid of the constant component but also of the common noise such as laser intensity fluctuations. For that, let’s consider a four-port fiber coupler like the one shown in Fig. 2.3. With the help of the matrix optics, a simple mathematical model regardless of the light coupling mechanism, the coupler can be described as a four-port device. For an ideal lossless 2 × 2 (50 : 50) coupler, the output is a unitary transformation of the input, which can be conveniently expressed in the form a unitary matrix [62]: " s3 s4 # = √1 2 " 1 j j 1 # " s1 s2 # (2.14)

Thus, the intensities (Eq. (2.4)) at the output ports are: i3 = 1 2|s1| 2₊1 2|s2| 2₊_<{s 1s∗2} (2.15) i4 = 1 2|s1| 2₊1 2|s2| 2_{− <{s} 1s∗2}. (2.16)

The differential use of two output ports produces a balanced detection. Each output port is aimed to a detector, and their output currents are subtracted. Since the first two unmodulated terms in Eqs. (2.15) and (2.16) are common to both detectors they cancel out in subtraction. In practice, however, the balancing is never perfect, but it does help rejecting unneeded common mode signals and noises. The signal of interest is also doubled. From a noise perspective, the standard deviation of noises that are independent in the two detection arms, such as thermal and shot noise will increase by a factor √2, but common mode noise such as light intensity fluctuations (RIN) will be subtracted. Balanced detection therefore pushes further the high

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C1 C2 (LO) π/2 D1 sQ ⊖ D2 sI ⊖

Figure 2.4: IQ detection scheme for a dual comb. The local oscillator (LO), in this case comb C2, passes through a phase shifter. This arm reads or detects in-quadrature signals sQ, while

the other one detects in-phase signals sI.

power SNR limitation imposed by source intensity fluctuations. Because common mode signal subtraction alleviates dynamic range constraints in the amplifying stages, balanced detection also enables sending more power to the photodetectors allowing further SNR improvements before non-linearity becomes a problem. Finally, in a balanced detection, the subtracted output can be either DC- or AC-coupled. With balanced detection, AC coupling provides an improvement with common mode signal rejection. The subtraction quality depends on the detector match. Thus, any mismatch will cause a DC offset current that can be minimized with an AC coupled amplifier.

Thus far, we have moved the limit in comb power, from the amplifier saturation for a DC-coupled unbalanced detection scheme towards the nonlinearity of the detector for an AC-coupled balanced one. In a next section, we will see that with a modification of the comb signal, this limit can be pushed even further.

Finally, we also consider another type of detection that has been extensively studied and, which is more commonly implemented in optical communication systems. In-phase and quadrature (IQ) detection measures the beat note of the signal not only with the cosine signal of a local oscillator but also with a 90-degree phase shifted or sine version of the local oscillator. This thus produces two beat notes at the frequency difference having the form cos[2π(f2− f1)t]

and sin[2π(f2− f1)t]. Combining them as cos θ + j sin θ allows retrieving the complex analytic

representation of the beating signal. Fig. 2.4shows an IQ detection scheme for a dual comb system. Among the advantages of this detection is the non-ambiguity in determining the relative positions of comb lines. For instance, let’s detect a pair of interfering modes with frequencies f2 and f1. The beat frequency will be f2− f1. In a direct detection, the retrieved

signal is a cosine whose Fourier transform gives two impulses at ±(f2− f1):

F{cos[2π(f2− f1)t]} = δ (f− (f2− f1))

2 +

δ (f + (f2− f1))

2 . (2.17)

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ν C2 C1 Direct IQ _F{C} F{R} f fr 2 −fr 2 f fr 2 −fr 2

Figure 2.5: IQ detection, a non-ambiguity measurement. A direct detection implies a real signal sI with a symmetric spectrum about f = 0. This symmetry means that it is not

possible to determine the relative position of the combs. On the other hand, an IQ detection allows composing a complex signal sI+ jsQ whose spectrum overcomes the ambiguity in the

combs position.

using Euler’s identity:

F{cos[2π(f2− f1)] + j sin[2π(f2− f1)]} =

δ (f _{− (f}2− f1))

2 . (2.18)

Now, considering that a sampled IGM is the set of many beats, IQ detection removes the redundant spectrum copy in the interval kfr±f₂r ∀k ∈ Z, see Fig.2.5. Though this is true only

to a certain extent, i.e. neither balanced detection is perfect nor the ratio and phase between the I and Q channels are ideal, IQ detection never quite perfectly cancels the negative copies. Nevertheless, IQ parameters can often be retrieved and an a posteriori correction can better cancel the negative aliases.

Thus far, we have given a context for noise and defined our signal and how it is measured. Now, we will present a modification over the pulses that can improve significantly the detection performance and its SNR. It is about chirping the combs.

2.4 Chirping

Chirp is defined as the time dependence of an optical pulse’s instantaneous frequency [63]. Diverse methods exist for generating chirped pulses, as frequency-shifted feedback lasers or Fourier domain mode-locked lasers [64], however for ultra-short pulses, the simplest and most straightforward way to implement chirp is to propagate pulses through a dispersive medium. Depending on the required chirp, several meters or even a spool of single mode fiber - SMF may suffice. However, for larger chirps, it may be better to use a fiber Bragg grating - FBG. For a given spectrum, the pulse temporal width is minimum when there is no chirp. As the energy is conserved, the chirp process implies lower pulse amplitudes. Furthermore, as chirping is a linear time-invariant operation the process is reversible, i.e. chirped pulses that are propagated in suitable conditions of dispersion can recover the initial conditions in width

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t i(µA) 10 5 76 fs 2.7 ps 27 ps i(µA) 1000 100 10 t 10 µA 42 µA 130 µA

Figure 2.6: Improving dynamic range by chirping. As one comb is chirped, the maximal IGM amplitude is reduced and the IGM is spread to a longer duration, allowing more comb power for using a given dynamic range. In the top axis, the left IGM comes from pulses without chirp with a temporal width of 76 fs. In the middle, the IGM comes form one pulse without chirp and the other with a temporal width of 2.7 ps. In the right IGM, the chirped pulse has a temporal width of 27 ps. In this figure, the power of chirped combs was increased to reach the same detected peak, in this case, 10 µA. At the bottom axis, IGMs for the last two graphs show their phase-corrected or unchirped versions. Note the log scale for the detected current.

and amplitude provided there are no losses. These features have been profited in dual comb spectroscopy.

According to Genest et al. [65], a differential chirp in the combs would make better use of the amplifier dynamic range. In fact, as differentially chirped combs produce chirped sampled IGMs, impulse response heights will be lower so that the requirements for the amplifier dynamic range are less stringent. Or this advantage can be used in another way, for a given dynamic range, it is possible to send more power to detectors with a larger chirp. This additional power allows combs to better illuminate the object under test; to get away from the detector noise floor, and to improve the dynamic range, see Fig. 2.6. The last two become evident when the chirp is removed via a software post-correction and the IGM gets the original amplitude and minimum temporal width. In fact, the detection chain can be seen as a dynamic range bottleneck. Before, the optical signal can have a very large dynamic range and the same apply for the digital signal after acquisition. Chirping is therefore reducing the signal’s dynamic range just before it is severely constrained. The linearity of the chirping operation

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ensures that the full dynamic range can be retrieved numerically afterwards.

Mathematically speaking, a second-order chirped pulse can be expressed rewriting Eq. (2.6) as:

˜

g1(t) = ˜a1(t− τ1) exp (jβc1t

2_{) exp [j2πν}

c(t− τ1)] exp (jφ1) (2.19)

with βc1 being the chirp coefficient. The correlation term in Eq. (2.12) will be:

Ck(∆τk, ∆φk) = exp (j2πνc∆τk) exp (j∆φk) · Z ∞ −∞ ˜ a1(t)˜a∗2(t + ∆τk) exp[jβc1(t) 2_{− jβ} c2(t + ∆τk) 2_]dt _(2.20)

One shall note that the differential chirp between the two frequency combs is the parameter of interest, much like the differential dispersion between the two arms is the key parameter for interferogram chirp in a conventional two beam interferometer.

2.5 Constant additive noise

We will now consider several types of noise in dual comb spectroscopy, starting with constant additive noise. This noise includes all effects of the acquisition system, such as thermal noise, amplifier noise, and acquisition card noise.

Let’s begin with the detector. The noise-equivalent power (NEP) expresses the sensitivity of the device and is given in watts per square root of hertz (W/√Hz) [66], indicating a power density rather than power. In this sense, the variance associated with the NEP should be defined in terms of a power spectral density (PSD) SNEP(f ) like:

σ2_NEP= Z f2

f1

SNEP(f )df. (2.21)

However, NEP is usually specified only by a single number indicating perhaps a flat PSD, in agreement with the fact that thermal noise is in theory white up to 1 THz. Thus, for a white noise PSD, the current noise variance in squared amperes is given by:

σ2_NEP=R2NEP2_B

H, (2.22)

where R is the detector responsivity, and the bandwidth BH over which noise is integrated by

the detection chain is called the equivalent noise bandwidth and is commonly defined as [67]:

BH = 1 2π Z ∞ −∞|H(jω)| 2_dω 2_|H(jω)|2 max , (2.23)

where H(jω) is the Fourier transform of detector’s impulse response. Electronic amplifiers also contribute in adding noise. Many detectors have an integrated amplification chain and