Homodyne array detection

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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