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## Image detection in the transverse plane

We present here the optical detection devices that are used throughout the experimental demonstrations of this thesis. The ¯rst one is the common photo-detector – also termed « bucket detector » – measuring the total beam power. Secondly, we will brie°y introducea measurement relying on the interference with another beam whose properties are well known, hence providing information on the mode quality of the incident beam. The third one is the homodyne detection, well adapted to the study of a particular spatial mode within an optical image, and which allows ¯eld quadrature measurement. The fourth one is the array detector, adapted to a pixel description of the transverse plane. Finally, we will investigate brie°y for the possibility of combining the two latter detections devices. All detection devices are presented in Fig. 1.10.

### Homodyne array detection

A homodyne array detection consists of a combination of identical array detectors in a homodyne detection con¯guration, as shown in Fig. 1.10e). Such a device has never been implemented experimentally, but could be very interesting as it would combine advantages of both array and homodyne schemes. Indeed, the pixel detectors allows simultaneous spa- tial measurements in the image, without having to change the local oscillator between each measurement, and the homodyne con¯guration allows to probe all di®erent quadratures of the incident image [Raymer93]. The shape of the local oscillator and the pixel gain distribution best adapted for a particular type of measurement will be investigated .

#### Spatial optical information carried by transverse modes

In this section, we detail the physical signi¯cance of the ¯rst Hermite Gauss modes relative to a bright TEM00 beam. A part of the work presented here has been published during the course of this PhD in reference [Hsu04]. We notably show that displacement and tilt, waist-size and waist-position mismatch of a TEM00 mode can be simply expressed in terms of Hermite Gauss modes. The corresponding modi¯cations of the TEM00 reference beam are represented in Fig.3.1. Finally, we introduce the angular momentum of the beam and will use a description of the ¯eld with Laguerre-Gauss modes. We limit our analysis to a one-dimensional description of the physical parameters, along the x axis. Nevertheless, an identical set of variables could be similarly de¯ned along the other orthogonal direction of the transverse plane, namely along the y axis.

**Rotation of a Laguerre Gauss beam about its propagation axis**

We propose here to introduce another orthonormal basis of the transverse plane, more ap- propriate20 for the description of beam rotation about the propagation axis, the Laguerre- Gauss (L-G) mode basis [Siegman86]. A Laguerre-Gaussian (LG) laser beam has an optical vortex in its center, i.e. that its phase representation is twisted like a corkscrew around the propagation axis. We call topological charge the integer number of twists the light does in one wavelength. Optical vortices are for instance used in optical tweezers to manipulate micrometer-sized particles such as cells. Such particles can be rotated in orbits around the axis of an orbital vortex [Paterson01]. Micro-motors have also been created using optical vortex tweezers [Luo00]. Laguerre-Gaussian modes can be generated experimentally with spiral phase plates [Beijersbergen94], or computer-generated holograms [Basistiy93].

**Quantum limits for information extraction from an optical image**

Before considering the general case of the optimal detection of a parameter within an optical image, let us ¯rst study the simple case of displacement and tilt measurement of a Gaussian beam. We will come back on more practical aspects of such measurements in section 5 A. A part of the work presented here has been published in reference [Hsu04].

**Table of contents :**

**1 Tools for Quantum Imaging **

**A Generalities on optical images**

A.1 Image description in the transverse plane

A.1.1 Local electric ¯eld operator

A.1.2 Local quadrature operators

A.1.3 Local number of photons and local intensity

A.1.4 Total number of photons and beam power

A.2 Transverse modal decomposition of the electromagnetic ¯eld

A.2.1 Modal creation and annihilation operators

A.2.2 Modal quadratures

A.2.3 Number of photons and intensity in a mode

A.2.4 Description relative to the mean ¯eld

A.2.5 Changing the transverse basis

A.2.6 The Hermite-Gauss basis

A.3 Gaussian quantum states of light

A.3.1 Covariance matrix

A.3.2 Coherent states

A.3.3 Squeezed states

A.3.4 Entangled states

**B Single/Multi-mode criterium **

B.1 Classical approach

B.2 Quantum approach

B.2.1 Single Mode quantum light

B.2.2 Multi-mode quantum light

B.3 Towards an experimental criterium

**C Image detection in the transverse plane **

C.1 Noise-modes of detection

C.2 « Bucket » detection

C.3 Interference detection

C.4 Homodyne Detection

C.5 Array Detection

C.5.1 Measured signal

C.5.2 Di®erence measurements

C.5.3 General linear measurement

C.6 Homodyne array detection

**D Conclusion **

**2 Quantum study of optical storage **

Overview on optical data storage

Article 1 : Optical storage of high-density information beyond the di®raction limit: a quantum study

Article 2 : A quantum study of multibit phase coding for optical storage

**3 Optimal information extraction from an optical image **

**A Spatial optical information carried by transverse modes **

A.1 Displacement and Tilt of Gaussian beams

A.1.1 Classical description

A.1.2 Quantum operators : position and momentum

A.1.3 Displacement and tilt of other beams

A.2 Waist position and size mismatch

A.2.1 Waist-size mismatch

A.2.2 Waist-position mismatch

A.2.3 General relation for mode-mismatch

A.3 Orbital angular momentum

A.3.1 Rotation of a Hermite Gauss beam about its propagation axis

A.3.2 Rotation of a Laguerre Gauss beam about its propagation axis

**B Quantum limits for information extraction from an optical image **

B.1 Displacement and tilt measurements

B.1.1 Quantum limits for displacement and tilt measurements

B.1.2 Optimal displacement measurements

B.1.3 Displacement and tilt measurements beyond the QNL

B.2 Quantum limits in general image processing

B.2.1 Introduction

B.2.2 Intensity measurements

B.2.3 Field measurements

B.2.4 Comparison

**C Conclusion **

**4 Transverse modes manipulation **

**A Basic manipulations of Hermite Gauss modes**

A.1 Propagation of Hermite Gauss modes

A.1.1 Gouy phase shift

A.1.2 Imaging in terms of Hermite Gauss modes

A.2 Generation of higher order modes

A.2.1 « Universal » mode-conversion devices

A.2.2 Hermite Gauss mode generation using a misaligned optical cavity

A.2.3 Displacement and tilt modulators

A.3 Combination of higher order modes

A.3.1 Beam-splitter

A.3.2 Special Mach-Zehnder

A.3.3 Ring cavity

**B Second Harmonic Generation with higher order Hermite Gauss modes **

B.1 Single pass SHG experiment

B.2 Thin crystal approximation

B.2.1 Transverse pro¯le of the generated SHG modes

B.2.2 Conversion e±ciency

B.3 Beyond the thin crystal approximation

B.3.1 Generalization of Boyd and Kleinman’s approach to higher order modes

B.3.2 Sensitivity to experimental parameters

B.4 Potential applications

**C Generation of higher order Hermite Gauss modes squeezing **

C.1 Theoretical analysis of TEMn0 mode Optical Parametric Ampli¯cation

C.1.1 Introduction

C.1.2 Multi-mode description of the parametric interaction

C.2 Experimental demonstration of higher order Hermite Gauss mode squeezing

C.2.1 Experimental setup

C.2.2 Optimization of the pump pro¯le

C.2.3 Optimization of the phase matching condition

C.2.4 TEM00, TEM10 and TEM20 squeezing

**D Conclusion **

**5 Quantum Imaging with a small number of transverse modes **

**A Experimental demonstration of optimal small displacement and tilt measurements**

A.1 Displacement and tilt measurements

A.1.1 Split-detection

A.1.2 Homodyne detection with a TEM10 mode local oscillator .

A.2 Displacement measurement beyond the standard quantum noise limit

A.3 Comparison of TEM10 homodyne and split-detection for displace- ment and tilt measurements

**B Spatial entanglement **

B.1 Theory

B.1.1 Heisenberg inequality relation

B.1.2 Entanglement scheme

B.1.3 Inseparability criterion

B.2 Experimental setup

B.2.1 Optical layout

B.2.2 Electronic layout

B.3 Experimental results

B.3.1 TEM10 mode quadrature entanglement

B.3.2 Towards spatial entanglement

**C Conclusion **

**Conclusion and Perspectives **

**Appendix **

A Array detection: two-zone case

A.1 Gain optimization

A.2 Non-di®erential measurement

**B Boyd-Kleinman’s derivation of SHG with higher order Hermite Gauss modes**

B.1 Calculation for the TEM00 pump mode

B.2 Calculation for the TEM10 pump mode

B.3 Calculation for the TEM20 pump mode

**C Knife-edge experiment for single and bi-mode ¯elds **

**Bibliography **