Mathematical Devices for Optical Sciences

Description

Modern optics is the physics of two-by-two matrices and harmonic oscillators. The one- and two-photon coherent states are based on the mathematics of harmonic oscillators. This comprehensive book examines the mathematical details of how two-by-two matrices, Wigner functions and the Lorentz group can be implemented in classical and quantum optics.   PRELIMS.pdf Preface CH001.pdf Chapter 1 Forms of quantum mechanics 1.1 The Schrödinger and Heisenberg pictures 1.2 Interaction picture 1.3 Density-matrix formulation of quantum mechanics 1.3.1 Mixed states 1.3.2 Density matrix and ensemble average 1.3.3 Time dependence of the density matrix 1.4 Further contents of Heisenberg’s commutation relations 1.4.1 Rotation group and its extension to the Lorentz group 1.4.2 Harmonic oscillators and Fock space 1.4.3 Dirac’s two-oscillator system References CH002.pdf Chapter 2 Lorentz group and its representations 2.1 Lie algebra of the Lorentz group 2.2 Two-by-two representation of the Lorentz group 2.3 Four-vectors in the two-by-two representation 2.4 Transformation properties in the two-by-two representation 2.5 Subgroups of the Lorentz group 2.6 Decompositions of the Sp(2) matrices 2.6.1 Bargmann decomposition 2.6.2 Iwasawa decomposition 2.7 Bilinear conformal representation of the Lorentz group References CH003.pdf Chapter 3 Internal space–time symmetries 3.1 Wigner’s little groups 3.1.1 O(3)-like little group for massive particles 3.1.2 E(2)-like little group for massless particles 3.1.3 O(2,1)-like little group for imaginary-mass particles 3.2 Little groups in the light-cone coordinate system 3.3 Two-by-two representation of the little groups 3.4 One expression with three branches 3.5 Classical damped oscillators References CH004.pdf Chapter 4 Photons and neutrinos in the relativistic world of Maxwell and Wigner 4.1 The Lorentz group and Wigner’s little groups 4.2 Massive and massless particles 4.3 Polarization of massless neutrinos 4.3.1 Dirac spinors and massless particles 4.4 Scalars, vectors, tensors, and the polarization of photons 4.4.1 Four-vectors 4.4.2 Second-rank tensor 4.4.3 Higher spins References CH005.pdf Chapter 5 Wigner functions 5.1 Basic properties of the Wigner phase-space distribution function 5.2 Time dependence of the Wigner function 5.3 Wave packet spread 5.4 Harmonic oscillators 5.5 Minimum uncertainty in phase space 5.6 Density matrix 5.7 Measurable quantities References CH006.pdf Chapter 6 Coherent states of light 6.1 Phase-number uncertainty relation 6.2 Baker–Campbell–Hausdorff relation 6.3 Coherent states 6.4 Symmetry of coherent states 6.5 Coherent states in phase space 6.6 Single-mode squeezed states References CH007.pdf Chapter 7 Squeezed states and their symmetries 7.1 Two-mode states 7.2 Unitary transformations 7.3 Symmetries of two-mode states 7.4 Dirac matrices and O(3,3) symmetry 7.5 Symmetries in phase space 7.6 Two coupled oscillators References CH008.pdf Chapter 8 Entanglement and entropy 8.1 Density matrix and entropy 8.2 Two-by-two density matrices 8.3 Density matrix for two-oscillator states 8.4 Entropy for the two-mode state 8.5 Entangled excited states 8.6 Wigner functions and uncertainty relations References CH009.pdf Chapter 9 Ray optics and optical activities 9.1 Ray optics using the group of ABCD matrices 9.1.1 Diagonalization properties of the ABCD matrices 9.1.2 Decompositions of the ABCD matrices 9.1.3 Recomposition of the ABCD matrices 9.2 Physical examples using ABCD matrices 9.2.1 Optics using multilayers 9.2.2 Ray optics applied to cameras 9.3 Optical activities 9.3.1 Computation of the transformation matrix U 9.3.2 Correspondence to space–time symmetries References CH010.pdf Chapter 10 Polarization optics 10.1 Jones vector, phase shifters, and attenuators 10.1.1 Squeeze and phase shift 10.1.2 Rotation of the polarization axes and combined effects 10.1.3 The SL(2,c) content of polarization optics 10.2 New filters and possible applications 10.3 Non-orthogonal coordinate systems References CH011.pdf Chapter 11 Stokes parameters and Poincaré sphere 11.1 Polarization optics and decoherence 11.2 Coherency matrix and Stokes parameters 11.3 Poincaré sphere 11.3.1 Two concentric Poincaré spheres 11.3.2 O(3, 2) symmetry of the Poincaré sphere 11.3.3 The Poincaré circle 11.3.4 Diagonalization of the coherency matrix 11.4 The entropy problem 11.5 Further symmetries from the Poincaré sphere 11.5.1 Momentum four-vector and the Poincaré sphere 11.5.2 Mass variation within O(3, 2) symmetry References APP1.pdf Chapter A.1 The covariant harmonic oscillator A.1.1 Differential equations of the covariant harmonic oscillator A.1.2 Normalizable solutions of the relativistic oscillator equations A.1.3 Lorentz transformations of harmonic oscillator wave functions A.1.4 Covariant phase-space picture of harmonic oscillators A.2 Quark–parton puzzle A.2.1 Lorentz-covariant quark model A.2.2 Feynman’s parton picture A.2.3 Proton structure function and form factor A.2.4 Coherence in momentum–energy space A.2.5 Hadronic temperature References INDEX.pdf Index