Mathematical Devices for Optical Sciences

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