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## The classical electromagnetic eld

Light being a form of electromagnetic radiation, its description may be achieved by Maxwell’s equations. Throughout this manuscript, every boldface symbol X denotes a vector in the carte-sian basis, unless speci ed otherwise.

**Description of the real electromagnetic eld**

We begin by writing the electric eld E(r, t), which is a 3-dimensional vector that depends on the spatial variable r and the temporal variable t.

In order to keep this description general, we consider that the eld propagates through a medium of free charge density ‰ and of polarization density P, neglecting the magnetic part. The electric eld will induce on matter an electric response D, called the electric ux density, which is de ned as D(r, t) ˘ « 0E(r, t) ¯P(r, t) (1.1)

More generally, the relationship between the applied electric eld E and the response D is established through the electric permittivity tensor « r: D(r, t) ˘ « 0 [« r]E(r, t) (1.2)

The physics behind the eld-matter interaction is then contained within the « r tensor, which describes the anisotropy of the medium. Its de nition will be particulary useful for the descrip-tion of non-linear e ects that will be outlined in chapter 8.

For now, we specialize to the case of propagation through a charge free ‰ ˘ 0, isotropic and linear medium. This involves that the relation between the induced polarization and the applied eld is linear: P(r, t) ˘ « 0´eE(r, t) (1.3)

Under these conditions, the relation between the electric eld and the response of the medium is simply given by D(r, t) ˘ « 0 « E(r, t) with » ˘ 1 ¯´e (1.4)

This lead to the de nition of the index of refraction n, which is more commonly used in optics: p (1.5) n ˘ »

The familiar propagation equation that governs the spatial and temporal propagation of the electric eld through a medium is: 4E ˘ v’2 @t2

where 4 stands for the vectorial laplacian operator and v’ ˘ c/n is the phase velocity, i.e. the speed of light in the medium (in the case of vacuum, we have naturally v’ ˘ c).

A standard solution to (1.6) is the plane-wave solution: E(r, t) ˘ RenE0 ei(k¢r¡!t¯`)o (1.7) where E0 is a constant vector, k is the propagation vector whose magnitude k ˘ !n/c satis es the dispersion relation for a plane-wave of pulsation !. In this expression, an arbitrary phase ` will be expanded in more detail in section 2.2.

**Fourier space formalism**

On many occasions in this manuscript, it will be convenient to look at the representation of the electric eld in the frequency-domain, which we shall describe in this part.

In this work, we will adopt the symmetric de nition of the Fourier transform. Although not necessary, it is convenient to use this prescription in quantum optics with continuous variables as the commutation relations for the bosonic operators aˆ(t) and aˆ(!) are then symmetric (see section 1.4.2).

For a function f (t) de ned in the temporal domain, we write the Fourier transform fe(!) de ned in the conjugated space as: Z dt (1.8) f (t) ei!t • F [ f (t)]

Conversely, the inverse Fourier transform is then given by1: f (!) e¡i!t • F¡1 [ f (!)] f (t) ˘ p

Applying this to the real electric eld yield its Fourier decomposition Z d! (1.10) E(r,!) e¡i!t E(r, t) ˘ p

2… Since E(t) is a real quantity, it follows that [E(r,!)]⁄ ˘ E(r,¡!) (1.11)

This de nition of E(!) therefore contains some redundancy, which leads to the introduction of the analytic electric eld E(¯) (r, t) where the negative frequencies are removed from the Fourier decomposition:

E(¯) (r, t) ˘RZ !

d E(r,!) e¡i!t (1.12)

It is worth stressing that this quantity is now complex, so that the real eld is de ned by the relation E(r, t) ˘ E(¯) (r, t) ¯E(¡) (r, t) (1.13) where E(¡) (r, t) ˘ £E(¯) (r, t)⁄⁄ corresponds to the integration over the negative frequencies.

Equivalently, one may de ne an analytic signal in the frequency domain by taking the Fourier transform of the temporal analytic signal: E(¯) (r,!) ˘ dt E(¯) (r, t) ei!t (1.14)

It follows that E(r,!) ˘ E(¯) (r,!) ¯E(¡) (r,¡!) (1.15) where E(¡) (r,!) ˘ £E(¯) (r,!)⁄⁄.

### Modal description

As introduced by equation (1.7), plane-waves satisfying the dispersion relation form a basis on which the eld can be expanded. More generally, it may be expanded on any set of normalized modes, either spatial, temporal, or spatiotemporal, as long as they satisfy Maxwell’s equations.

In this section, we show how to describe the electric eld with modes in the longitudinal and transverse plane. We enclose the system in a box of volume V and of section S.

**Temporal and spectral modes**

A decomposition of the eld in plane-waves may be achieved by expanding the analytic eld (1.12) in spatial Fourier components, as it is done in [Grynberg 10]. The eld is then written as E(¯) (r, t) ˘ i XE‘ fi‘ « ‘ ei(k‘¢r¡!‘ t) (1.16) P‘ fi‘ u‘(t) is the where « ‘ is the polarization of the component ‘, k‘ its wavevector, fi‘ is the normal variable which corresponds to the complex amplitude of the component ‘, and E‘ is a normalization constant given by E‘ ˘ s (1.17) 2nc »0V

This is called the normal mode decomposition and each mode of the basis is an independent monochromatic polarized wave. This de nition of the eld is very convenient when quanti-fying it, but for the scope of this thesis, we will rather decompose a light beam on a basis of envelope modes. For the remaining of this manuscript, we will consider the eld only in a given linear polar-ization, E is then reduced to a scalar. We also consider that the frequency spectrum in (1.16) is narrow and centered around !0, allowing the constant to be taken out of the sum E‘ ’ E0. Finally, we rewrite (1.16) as a decomposition of envelope modes u(t) relative to the carrier frequency: E(¯) (r, t) ˘ E0 Xfi‘ u‘(t)ei(k¢r¡!0 t)(1.18) where {u‘(t)} is a set of orthonormal modes that satisfy the general condition (1.31) and fi‘ is the complex amplitude of the eld. We’ve also incorporated the imaginary unit i in the mode u‘, since these can always be de ned up to a constant phase factor. It will sometimes be convenient to write the eld as E(¯) (r, t) ˘ E0a(t)ei(k¢r¡!0 t) where a(t) • envelope of the eld.

By taking the Fourier transform of (1.18), one may also de ne a spectral mode, or frequency mode: E(¯) (r,!) ˘ E0 Xfi‘ u‘(! ¡!0) eik¢r with u(!¡!0) ˘ u(›) ˘ F [u(t)] and › ˘ !¡!0 is the frequency relative to the optical carrier.

These temporal – or spectral – modes will be the main center of focus throughout this the-sis. Their de nition is very general at this point since the modes {u‘} needs only to satisfy Maxwell’s equation as well as the normalization and orthogonality conditions (1.31). How-ever, in section 2.1.4, we will revise this spectro-temporal modes concept by applying it to the case of ultrashort laser pulses. In particular, we will use whenever possible the gaussian pro le for the spectral and temporal envelopes, as every calculation will have an analytical solution in this case.

**Spatial modes**

The previous treatment only deals with plane waves whose wavefront is in nite. However, in practice, actual laser beams have a nite transverse extent and may not be considered as true plane waves.

Fortunately, in the present case, we may consider the laser beams as paraxial, meaning that they are made up of a superposition of plane waves with propagation vectors close to a single direction. This also implies that the eld’s variations in the transverse plane are much slower than in the longitudinal dimension.

We choose the propagation direction as z, and the transverse direction as the (x, y) plane where we de ne a unitary vector ‰. Therefore, the position vector is written as r ˘ ¡‰, z¢. A more complete description of the paraxial beams and the transverse structure of laser eld may be found in [Yariv 67] or [Siegman 86].

We consider a monochromatic paraxial wave written as E(¯) (r, t) ˘ E0 g(r) ei(kz¡!0 t) (1.20) where E0 ˘ E0fi encompasses the eld amplitude, k satis es the dispersion relation and g is a slowly varying envelope in the longitudinal direction. Mathematically, this condition is written flfl@2z gflfl¿ 2k j@z gj and allows to the neglect second order derivatives of g with respect to z.

Injecting the expression (1.20) into the propagation equation (1.6) under this approximation leads to the following paraxial wave equation 4‰ g ¡2ik @g (1.21) @z ˘ 0 where 4‰ ˘ @2x ¯@2y is the laplacian operator in the transverse plane.

This equation has gaussian solutions that provide a good description of the laser beams that we are used to work with. In particular, the entire family of transverse electromagnetic mode (TEM) prove very useful as they correspond to the spatial eigenmodes of a laser cavity.

#### Spatio-temporal modes

The previous de nitions in the transverse and longitudinal domains are quite convenient, since they may be combined in a straightforward manner to build a new set of modes. This provides a complete model description of the electric eld.

Under the previous descriptions and approximations, a linearly polarized electric eld may be expanded on the basis of temporal ui(t) and spatial modes vn(x, z) as: E(¯) (x, z, t) ˘ E0 Xfii,nui(t) gn(x, z) ei(kz¡!0 t) Alternatively, we are also able to de ne a new basis of modes wi,n(x, every combination of the longitudinal and transverse modes: wi,n(x, z, t) ˘ ui(t) gn(x, z) (1.29) z, t) that encompasses (1.30)

Note that the spatial and temporal parts are factorized in w, which assumes no space-time coupling. This is a very reasonable assumption for the present work, where the light beam is in a well-de ned spatial mode. At any position z and over a detection time T, these form an orthonormal set; introducing the standard L2 inner product h¢,¢i, it reads ›wi,m, wj,nfi• c dt d2‰ w⁄i,m wj,n ˘ ScT –i j –mn

**Basis change**

The modes that we chose, being the temporal u(t) or spatial v(x, z) modes, are not unique; the eld may be expanded on any other basis. As an example, if we consider another temporal basis {vi(t)} of the eld, the change from {u i(t)} to {vi(t)} is achieved by a unitary transform U

**The quadratures of the classical eld**

In section 1.2, we’ve seen that we can write the eld in a particular spatio-temporal mode as the product of a slowly varying envelope and a phase factor that re ect the wave-like nature of light. In the following parts, it will be useful to break this phase factor into an absolute phase and the wave front curvature part. This leads to the introduction of the eld quadratures [Bachor 04].

**Quadrature amplitudes**

Using the previous notations, we write the real electric eld in the spatio-temporal modes basis wi,n(x, z, t) as E (x, z, t) ˘ E0 Xfii,n wi,n(x, z, t) ei(kz¡!0 t) ¯c.c. • E0 a(x, z, t) e¡i!0 t ¯c.c. (1.43) where c.c. stands for conjugated complex, and where we merged the spatial propagation with the envelope to form the complex amplitudes a(x, z, t) ˘ Pi,n fii,n wi, n(x, z, t) eikz. An equiva-lent form of this notation is given in terms of the quadrature amplitudes X and P associated to the sine and cosine waves: E (x, z, t) ˘ E0 [X(x, z, t) cos(!0 t) ¯ P(x, z, t) sin(!0 t)] (1.44)

**Table of contents :**

**Introduction **

**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