Chapter 1

1.1

Introduction

4

1.2

Basic notations of scalar singular optics

4

1.2.1

Phase vortices

4

1.2.2

Topological charge and topological index of singular points. Elementary topological reactions

7

1.2.2.1

Topological charge

7

1.2.2.2

Topological index

8

1.2.2.3

Conservation law for topological charge

8

1.2.2.4

Elementary topological reactions

8

1.2.3

Experimental observation and identification of vortices into scalar fields

9

1.2.4

Generation of vortices using computer-generated holograms

10

1.3.

Vortices and phase structure of a scalar field

11

1.3.1

Sign principle

11

1.3.2

Phase speckles. “Breathing” of phase speckles

12

1.3.3

Birth of vortices

13

1.3.4

Appearance of wave front dislocations as a result of interference of waves with simple phase surfaces

14

1.3.5

Topological indices of the field of intensity. Extrema of phase and intensity. “Correlation” of phase and intensity

18

1.3.6.

Vortex nets. Phase skeleton of a scalar field

23

1.3.6.1

Reconstruction of the field’s phase on the basis of shifted vortex nets

23

1.3.6.2

Image reconstruction using a regular sampling found from analysis of the parameters of vortices in random field

26

1.3.6.3

Some remarks on the field reconstruction by the use of nets of the intensity stationary points

29

1.4.

Singularities of a vector field

31

1.4.1

Disclinations. Polarization singularities

31

1.4.2

Vortices of phase difference. Sign principle for a vector field

35

1.4.2.1

Field decomposition into orthogonally linearly polarized components

35

1.4.2.2

Principle of the vortex analysis of vector fields

36

1.4.2.3

Vortices of orthogonally polarized field components. Technique for study of polarization singularities

41

1.4.2.4

-points as the phase difference vortices

45

1.4.3

“Correlation” of intensity and polarization of the vector field

46

1.4.4

Iinterconnection of the component vortices and -points

48

1.4.5

Elementary polarization structures and elementary polarization singularities of vector fields

53

1.4.5.1

Polarization structures resulting from interference of orthogonally linearly polarized beams

53

1.4.5.2

Elementary polarization singularities resulting from interference of orthogonally circularly polarized beams

57

1.4.5.3

Experimental modeling of elementary polarization singularities

58

1.4.6

Fine structure and averaged polarization characteristics of inhomogeneous vector fields

60

1.4.6.1

Averaged Stokes parameters

61

1.4.6.2

Analysis of the averaged parameters for decomposition of the field into linearly polarized components

62

1.4.6.3

Computer simulation of the vector field’s parameters

67

1.4.6.4

Analysis of the averaged parameters for the field decomposition into circular basis

70

1.4.6.5

Comparison of the experimental results and the data of computer simulation

72

1.4.7

“Stokes-formalism” for polarization singularities. “Stokes-vortices”

74

1.5.

Singularities of the Poynting vector and the structure of optical fields

77

1.5.1

General assumption. Components of the Poynting vector

78

1.5.2

Singularities of the Poynting vector in scalar field

80

1.5.2.1

Instantaneous singularities of a scalar field

80

1.5.2.2

Averaged singularities of the Poynting vector of scalar field

83

1.5.3

Singularities of the Poynting vector at vector fields

85

1.5.3.1

Instantaneous singularities of vector field

85

1.5.3.2

Behavior of the Poynting vector in areas of elementary polarization singularities

88

1.5.3.2.1

Symmetric distributions of amplitude and phase of the interfering beams

88

а

Single -point

88

b

Angular momentum of the field into vicinity of С-point

90

c

Elementary polarization cells with two -points of the same signs

92

d

Elementary polarization cells with two -points of opposite signs

93

1.5.3.2.2

Non-symmetrical distributions of amplitudes and phases of the interfering beams

94

1.5.3.2.3

Experimental proving of the existence of the orbital momentum in the vicinity of -point

98

1.5.3.3

The averaged Poynting vector of the vector field

102

Appendix 1.1. Wave fronts approximation

104

Appendix 1.2. Fourier image of isotropic vortex

108

Appendix 1.3. Poynting vector. paraxial approximation

110

References for chapter 1

113