Noise analysis of an ultra-fast frequency comb 

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Representations of the pulse

From the previous section, it is clear that a full representation of the pulsed eld needs to include both the spectral and the temporal phases. Of course, once retrieved, one could plot the spectral and temporal phases of the pulse, but these can be hard to read. Although it does contain all the required information, it is not straightforward to know the temporal distribution of each frequency of the eld. For example, temporal representations of the pulse in the previous section do not give easily away the arrival time of each color.

Time-frequency distributions

In many other elds such as quantum mechanics or acoustics, other representations were introduced to complement standard Fourier analysis. These distributions of time-varying spectra are called spectrograms (or equivalently sonograms). The concept has been widely used for the analysis of time-varying spectra.
Over the years, a great number of distributions have been introduced and investigated, and it is still an evolving eld. Here we only expose the general principle that lies behind timefrequency distributions since it also hints as to how to access the pulse shape experimentally. It also provides a good visual witness of the pulse shape and allows one to understand rapidly the structure that the pulse acquired during its propagation. For a review on all the dierent time-frequency distributions, see [Cohen 89].

Wigner distribution

The whole eld of time-frequency distribution has been built upon the study of the Wigner function. First introduced by Wigner and applied to quantum mechanics, the Wigner function can be applied to any set of conjugated variables, for example, the wave-vector k and the position x, or the angular frequency ! and the time t. It has found some application in ultrafast optics for the description of pulses [Walmsley 96], but its usage has become limited since it is cumbersome to relate to the physical spectrum. Its mathematical denition is generally given by W (t,!) Æ Z R d¿ 2¼ E(¡) ³ t¡ ¿ 2 ´ e¡i!¿E(Å) ³ tÅ ¿ 2 ´.

Generation of pulses of light

In part III, we shall dive in more details into the noise characteristics of a laser source and we will need a decent knowledge on how the light of a femtosecond oscillator is generated. The aim of this section is to provide a description centered mainly on the type of lasers that have been used during this PhD (solid-state Titanium Sapphire laser, passive Kerr-lensing modelocking), and does not pretend to review all of the laser theory.

Steady-state laser cavity

For our purposes, we consider laser sources that are made of a linear Fabry-Pérot cavity with a gain medium, as depicted by gure 2.7. Light passes through the gain medium twice per round-trip, and the electric eld is periodic on this length. To achieve optical gain, a population inversion must occur in the gain medium. This corresponds to the situation where the number of electrons in an excited state exceeds the number of electrons in a lower level. This is usually achieved by optical pumping, where an external light source -e.g. a laser diode- is used to promote the electrons in an excited state.

Table of contents :

I Measuring with ultra-fast frequency combs 
1 The modes and states of a beam of light 
1.1 The classical electromagnetic eld
1.1.1 Description of the real electromagnetic eld
1.1.2 Fourier space formalism
1.2 Modal description
1.2.1 Temporal and spectral modes
1.2.2 Spatial modes
1.2.3 Spatio-temporal modes
1.2.4 Basis change
1.2.5 Power and energy
1.3 The quadratures of the classical eld
1.3.1 Quadrature amplitudes
1.3.2 Quadrature uctuations
1.4 Quantization of the eld
1.4.1 Bosonic operators
1.4.2 Modal decomposition
1.4.3 Quadrature operators
1.4.4 Relation to the classical eld
1.5 Quantum states
1.5.1 Density operator
1.5.2 Wigner function
1.6 Gaussian states
1.6.1 Denition and quantum covariance matrix
1.6.2 Examples of Gaussian states
2 Femtosecond ultrafast optics 
2.1 Description of pulses of light
2.1.1 Optical frequency combs
2.1.2 Energy and peak power
2.1.3 Moments of the eld
2.1.4 Gaussian pulses
2.2 The inuence of dispersion
2.2.1 Spectral and temporal phases
2.2.2 Eects on the pulse shape
2.3 Representations of the pulse
2.3.1 Time-frequency distributions
2.3.2 Some examples
2.3.3 Experimental realizations
2.4 Generation of pulses of light
2.4.1 Steady-state laser cavity
2.4.2 Mode-locked lasers
3 Revealing the multimode structure 
3.1 General experimental scheme
3.1.1 Laser source
3.1.2 Interferometric photodetection
3.1.3 Pulse shaping
3.2 Signal measurement
3.2.1 Modulations of the eld
3.2.2 Data acquisition
3.3 Mode-dependent detection
3.3.1 Quantum derivation
3.3.2 Spectrally-resolved homodyne detection
3.3.3 Temporally-resolved homodyne detection
3.3.4 Addendum: single diode homodyne detection
II Quantum metrology 
4 Parameter estimation at the quantum limit 
4.1 Projective measurements
4.1.1 Displacements of the eld in specic modes
4.1.2 Sensitivity
4.1.3 The Cramér-Rao bound
4.2 Spectral and temporal displacements
4.2.1 Temporal displacements
4.2.2 Spectral displacements
4.2.3 Conjugated parameters
4.2.4 Application to range-nding
4.3 Space-time coupling: a source of contamination
4.3.1 Transverse displacements
4.3.2 Homodyne contamination
5 Measuring the multimode eld 
5.1 Experimental details
5.1.1 Measurement strategy
5.1.2 Phase modulation at high frequencies
5.1.3 Spatial ltering
5.2 Interferometer calibration
5.2.1 Calibration of displacement
5.2.2 Sensitivity measurement
5.3 Multipixel detection
5.3.1 Design and construction
5.3.2 Gain calibration
5.3.3 Space-wavelength mapping
5.3.4 Clearance
5.4 Spectrally-resolved multimode parameter estimation
5.4.1 A glimpse at the multimode structure
5.4.2 Signal extraction
5.4.3 Heterodyne measurements: the need for a stable reference
5.4.4 Space-time positioning
5.4.5 Dispersion
5.4.6 Quantum spectrometer
III Noise analysis of an ultra-fast frequency comb 
6 Optical cavities 
6.1 Fabry-Perot cavities
6.1.1 Input-output relations
6.1.2 Characteristic quantities
6.1.3 Spatial mode
6.1.4 Noise ltering
6.1.5 Quadrature conversion
6.2 Synchronous cavities
6.2.1 Resonance condition
6.2.2 The cavity’s comb
6.2.3 Simulations
6.3 Experimental realization
6.3.1 Motivations
6.3.2 Design and construction
6.3.3 Cavity lock
6.3.4 Environnemental pressure dependency
6.3.5 Noise properties
7 Experimental study of correlations in spectral noise 
7.1 The modal structure of noise
7.1.1 Introduction and motivations
7.1.2 The noise modes
7.2 Measuring spectral correlations in the noise
7.2.1 Classical covariance matrix
7.2.2 Retrieving the uctuations
7.2.3 Experimental scheme
7.3 Experimental results
7.3.1 Amplitude and phase spectral noise
7.3.2 The noise modes
7.3.3 Collective parameters projection
7.3.4 Phase-amplitude correlations
7.3.5 Real-time laser dynamics analysis
IV Going further with quantum frequency combs 
8 Multimode squeezed states 
8.1 Generating quantum states
8.1.1 Creation of squeezed states
8.1.2 Parametric down conversion with an optical frequency comb
8.1.3 Objectives and perspectives
8.2 Single-pass squeezing
8.2.1 Parametric down conversion
8.2.2 Eigenmodes of the parametric down conversion
8.2.3 Expected eciency
8.3 Second harmonic generation
8.3.1 Eciency
8.3.2 The inuence of temporal chirp
8.4 An ultra-fast squeezer
8.4.1 Pump generation
8.4.2 Synchronously pumped optical parametric amplier
8.5 Perspectives
8.5.1 Quantum enhanced metrology
8.5.2 Entanglement
Conclusion and outlooks
Appendix A Medium dispersion 
A.1 Sellmeyer equation
A.2 Wave-vector dispersion
A.3 Application to delay and dispersion estimation
Appendix B Projective measurements by pulse shaping 
B.1 Pulse shaping the time-of-ight mode
B.2 Locking on the time-of-ight mode
B.3 Dispersion measurement
Appendix C Experimental construction of the detection modes 
Appendix D Conjugated variable of space-time position 
D.1 Detection mode for a global displacement
D.2 Detection mode for a spectral displacement
D.3 Conjugated parameter .

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