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PARTIAL COHERENCE AND OPTICAL VORTICES by Ivan Maleev A Dissertaion Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Physics by

___________________ June 9th, 2004 APPROVED:

_______________________________________________ Dr. Grover A. Swartzlander, Jr. Dissertation Advisor _______________________________________________ Dr. Padmanabhan K. Aravind, Committee Member _______________________________________________ Dr. Germano S. Iannacchione

ABSTRACT Optical vortices are singularities in phase fronts of optical beams. They are characterized by a dark core in the center and by a helical wave front. Owing to azimuthal components of wave vectors, an optical vortex carries orbital angular momentum. Previously, optical vortices were studied only in coherent beams with a well-defined phase. The object of this dissertation is to explore vortices in partially coherent systems where statistics are required to quantify the phase. We consider parametric scattering of a vortex beam and a vortex placed on partially coherent beam. Optical coherence theory provides the mathematical apparatus in the form of the mutual coherence function describing the correlation properties of two points in a beam. Experimentally, the wave-front folding interferometer allows analysis of the cross-correlation function, which may be used to study partial coherence effects even when traditional interferometric techniques fail. We developed the theory of composite optical vortices, which can occur when two coherent beams are superimposed. We then reported the first experimental observation of vortices in a cross-correlation function (which we call spatial correlation vortices). We found numerically and experimentally how the varying transverse coherence length and position of a vortex in a beam may affect the position and existence of spatial correlation vortices. The results presented in this thesis offer a better understanding of the concept of phase in partially coherent light. The spatial correlation vortex presents a new tool to manipulate coherence properties of an optical beam.

To Inga

ACKNOWLEDGMENTS This PhD would not be possible without several individuals, few groups and a bit of luck. The foundation of what I know about physics and math come from my professors at the Moscow Institute of Physics and Technology. That basis was fortified and developed by my mentors at the Worcester Polytechnic Institute and at the Optical Sciences Center, University of Arizona. The theory is nothing without experiment. And so I’d like to thank my advisor, Dr. Grover A. Swartzlander, Jr., for guiding me through all those years. His thorough approach to physics is undoubtedly cast on me, from conducting experiments to presenting results, albeit I’ll probably never reach his “degree of precision”. I would also like to thank him for the patience and last but not least, financial support. I’m also grateful to the members of my Graduate Committee, Dr. Padmanabhan K. Aravindand and Germano S. Iannacchione for reviewing this work. My grateful acknowledgments go to WPI Physics department and its Head, Dr. Thomas H. Keil. They always provided me with financial and logistical support. However the best thing about WPI Physics, in my opinion, is the size of the department. I always felt a part of the community where people actually know and care about what you are doing. My gratitude also goes to the Optical Sciences Center, where I spent last 2.5 years, and its Director, Dr. James C. Wyant, who has always been extremely kind to me. Among the number of

iii people at OSC I’d like to especially acknowledge the help of Dr. Arvind Marathay, who taught me coherence theory and whose advises were truly invaluable on so many occasions. Over these years I’ve always been very fortunate to have fellow graduate students who I talked to on an everyday basis, exchanging and discussing ideas, brainstorming and gaining from their experience. I’d like to especially name here David Rozas, Anton Deykoon and David Palacios. Their contribution to this work is definitely not less than my own. Finally, I can say that this PhD took a lot of time. Lots of things changed over these years including the focus of the work. And so I can now acknowledge that it would never be finished without my wife, Inga. Thank you.

CONTENTS Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1 Optical vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2 Detection of Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.3. Partial coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4. Mutual coherence function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.5. Partially coherent sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.6. MCF investigation, vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.7. Propagation, beam spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.8. Spectral and temporal coherence effects . . . . . . . . . . . . . . . . . . . . . . . .

17

2.9. Interaction of Optical Vortices with non-linear crystals . . . . . . . . . . . .

18

3. Detection of Optical Vortices in Partially Coherent Light . . . . . . . . . . . . . . . . . .

22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.2 Coherent truncated beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.3 Center of mass calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

CONTENTS

v

3.4 Partially coherent half-beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.5 Geometrical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.6 Half-vortex experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

4. Composite Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.2 Single Optical Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.3 Composite Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.4 Vortex Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

5. Spatial Correlation Vortices: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.2 On- and off-axis vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.3 Mutual coherence function, near field . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.4 Far-field analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5.5 Numerical calculations of the far-field MCF . . . . . . . . . . . . . . . . . . . . . .

76

5.6 Analytical solution, Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6. Spatial Correlation Vortices: Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

6.2 Optical vortex and a phase mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

CONTENTS

vi

6.3 Wavefront-Folding Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

6.4 Experimental Observation of Spatial Correlation Vortices . . . . . . . . . . .

93

6.5 Geometrical Model Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

7. Spontaneous Down Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104

7.2 Optical Vortex and Orbital Angular Momentum . . . . . . . . . . . . . . . . . .

105

7.3 Phase-Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106

7.4 Signal spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108

7.5 Quantum and Wave Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

110

7.6 Numerical Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

Appendix

122

A. Fourier Transform of Laguerre-Gaussian beams . . . . . . . . . . . . . . . . . . . . . . . .

123

LIST OF FIGURES 2.1 Single optical vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.2 Notation relating to the propagation of mutual coherence function . . . . . . . . . . .

22

3.1 Vortex intensity and phase, and related notation . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.2 Numerical simulations of coherent half-beams . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.3 Numerical simulations of partially coherent half-beams . . . . . . . . . . . . . . . . . . .

38

3.4 Diagram of partially coherent vortex beam in x-space . . . . . . . . . . . . . . . . . . . .

39

3.5 Diagram of beam-spread conical projections in k-space . . . . . . . . . . . . . . . . . . .

40

3.6 Experimental setup for partially coherent truncated beams investigation . . . . . .

41

3.7 Experimental images of partially coherent half-beam . . . . . . . . . . . . . . . . . . . . .

42

3.8 Partially coherent half-beam, center-off-mass and core visibility . . . . . . . . . . .

43

4.1 Two beams placed symmetrically with respect to the origin . . . . . . . . . . . . . . .

58

4.2 Composite beams created by interference of two (+1)-charged vortex beams . . .

59

4.3 Trajectories of composite vortices created from two (+1)-charged vortex beams

60

4.4 Composite beams, created from two oppositely charged vortices . . . . . . . . . . . .

63

4.5 Trajectories of composite vortices created from two oppositely charged vortices

64

4.6 Composite beams, created from vortex and Gaussian beams . . . . . . . . . . . . . . . .

66

4.7 Trajectories of the composite vortices created from vortex and Gaussian beams

67

5.1 Intensity profile and the phase map of an off-axis optical vortex . . . . . . . . . . . . .

84

5.2 Correlation function of a pair of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5.3 Numerically calculated cross-correlation, on-axis vortex case . . . . . . . . . . . . . . .

86

5.4 Numerically calculated cross-correlation, off-axis case, varying displacement . .

87

5.5 Same as Fig. 5.4, varying coherence length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

LIST OF FIGURES

viii

5.6 Separation of spatial correlation vortices in cross-correlation . . . . . . . . . . . . . . .

89

5.7 Separation of spatial correlation vortices in normalized cross-correlation . . . . . .

90

6.1 A diagram of partially coherent light transmitted through a phase mask . . . . . . .

99

6.2 Cross-correlation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

6.3 Experimental setup for cross-correlation investigation . . . . . . . . . . . . . . . . . . . . . 101 6.4 Experimental interferograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.5 Distance between spatial correlation vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

7.1 Classical description of orbital angular momentum . . . . . . . . . . . . . . . . . . . . . . .

115

7.2 Spontaneous down-conversion in a non-linear crystal . . . . . . . . . . . . . . . . . . . . .

116

7.3 Profile of the conversion form-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

7.4 Part of the numerically calculated signal down-conversion ring . . . . . . . . . . . . .

118

7.5 Experimentally obtained spontaneous down-conversion signal and idler rings . .

119

7.6 Down-conversion ring profile, quantum picture . . . . . . . . . . . . . . . . . . . . . . . . . .

120

7.7 Down-conversion ring-profile, wave-picture . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

SYMBOLS AND ABBREVIATIONS OV

-

Optical Vortex

SCV

-

Spatial Correlation Vortex

Γ

-

Mutual Coherence Function (MCF)

r K

-

wavevector

r k

-

transverse wavevector

λ

-

wavelength

r, φ

-

polar coordinates in x-space

x, y

-

Cartesian coordinates in x-space

k⊥

-

transverse polar coordinate in k-space

kx , ky

-

Cartesian coordinates in k-space

π

-

3.141592654

E

-

electric field

E0

-

electric field amplitude

I

-

intensity

t

-

time

m

-

topological charge of an optical vortex

w

-

size of a generic beam

w0

-

size of a Gaussian beam

i

-

imaginary unit, i = − 1

1

1. INTRODUCTION

Vortices are fascinating topological features which are ubiquitous throughout the Universe. They may be found in many branches of physics ranging from superfluidity1 to aero- and hydro-dynamics2 and global weather patterns3. Over the last two decades, potential applications raised interest in vortices within optical fields4. Optical vortices (OVs) were found to occur in coherent radiation such as Laguerre-Gaussian laser beams5, light scattered from rough surfaces6, 7, optical caustics8, 9 and OV solitons10, 11. Possible applications of optical vortices include optical switching12, optical trapping13,

14

and

coherence filtering15, 16. OVs are phase objects and require a degree of correlation between different points in the beam. Coherent OVs are also known to carry an associated orbital angular momentum. This thesis analyzes OVs in the partially coherent light, where the statistical properties of light should be taken into account to quantify global phase properties17, 18. A brief introduction to major definitions and a historical account concerning optical vortices and partial coherence is given in Chapter 2. The partial coherence effects may be investigated in two ways. The main approach, which is the focus of this thesis, starts with the partially coherent light source. The OVs are then created by passing a partially coherent beam through a phase mask, and the resulting partially coherent vortex beam is investigated. The problem of detection of optical vortices in partially coherent light, where the traditional interferometric techniques may fail, is addressed in Chapter 3. We use a vortex beam, truncated at the

1. INTRODUCTION

2

waist4 and experimentally analyze the far-field distribution, where an asymmetrical intensity distribution is expected due to the helicity of a vortex wavefront. The presence of OVs may be expected to change the coherence properties of a source. The main contribution of this author and his collaborators is the investigation of optical singularities, which may form in the cross-correlation function of the resulting beam. For the first time we detect experimentally, and investigate theoretically, this new type of optical vortices, named spatial correlation vortices. Our ability to selectively change the spatial coherence properties of a light beam may potentially help to develop a new branch of applications for optical vortices. Examples include optical coherence tomography, where control of the spatial coherence can suppress the coherent scattering from the undesirable areas, and coherence filtering. The ground work is presented in Chapter 4, where we discuss composite optical vortices that may be created in the coherent regime when two collinear optical vortex beams are superimposed19. Mathematical modeling and numerical calculation results for the partial coherence case are presented in Chapter 5. Experimental investigation of spatial correlation vortices is presented and discussed in Chapter 6. The second approach to the study of partial coherence involves passing a coherent beam of interest (e. g. optical vortex) through an optical system which affects coherence properties of a beam, for example a diffuser20. Optical down-conversion, also involving the non-linear transformation of a pump photon into a signal and an idler may serve as an example. The statistics of the signal and the idler beams, produced by down-conversion of a vortex pump beam, are presented in Chapter 7.

1. INTRODUCTION

3

Parts of the thesis have been published in or submitted to the following journals:

5. I. D. Maleev, D. Palacios, A. Marathay, G. A. Swartzlander, “Spatial correlation vortices: Experiment”, in preparation for Opt. Lett.

4. I. D. Maleev, G. A. Swartzlander, “Propagation of spatial correlation vortices”, in preparation for Opt. Comm.

3. I. D. Maleev, D. Palacios, G. A. Swartzlander, “Spatial correlation vortices: Theory”, submitted to J. Opt. Soc. Am. B.

2. D. Palacios, I. Maleev, A. Marathay, G. A. Swartzlander, Jr., “Spatial Correlation Singularity of a Vortex Field”, Phys. Rev. Lett., 92, 143905 (2004)

1. I. D. Maleev, G. A. Swartzlander, “Composite optical vortices”, J. Opt. Soc. Am. B, 20, 1169 (2003).

and were presented at the following international conferences:

4. I. D. Maleev, G. A. Swartzlander, “Hunting for Optical Vortices”, Annual OSA Meeting (2003).

1. INTRODUCTION

4

3. I. D. Maleev, G. A. Swartzlander, “Down-conversion ring profile as a function of the vorticity in a pump beam”, Annual OSA Meeting (2001).

2. I. D. Maleev, A. M. Deykoon, G. A. Swartzlander, Jr., “Transformation of topological charge in optical down conversion”, QELS-2000, Tech.-Digest., 40, 218-19 (2000).

1. I. D. Maleev, A. M. Deykoon, G. A. Swartzlander, Jr., M. S. Soskin, A. V. Sergienko, “Violation of conservation of topological charge in optical down conversion”, QELS1999, Tech.-Digest., 99 (1999).

1-20

2. HISTORY

2.1 Optical vortex Optical vortices (OV’s) occur as particular solutions to the wave equation in cylindrical coordinates21. The essential characteristic of a field vortex is the helical phase front. For a single vortex (see Fig. 2.1), centered on the optical axis, the transverse phase profile is given by Φ (r ,φ ) = mφ , where (r , φ ) are the transverse polar coordinates, and m is a

signed integer called the “topological charge”. In the following description we assume the non-trivial case m ≠ 0. The center of a vortex for a coherent beam is characterized by a dark core [for m=1 see Fig. 2.1(a)], within which the intensity vanishes at a points15, 16. As one travels a full circle around the core, the phase of the electric field changes by an integer factor of 2π (measured in radians), as shown in Fig. 2.1(b). At the center of the core the phase is undefined; this is physically recognizable since the intensity vanishes at this point. For this reason, a vortex is sometimes called a phase singularity. Assuming the path of the vortex beam lies on the optical axis, z, and adding the initial phase β of a field, which carries a vortex, and a phase factor, -kz, to describe the light propagating paraxially along the z-axis, using cylindrical coordinates (r , φ , z ) , we obtain the phase Φ (r ,φ , z ) = mφ + β − kz , where k is the wavenumber. In three dimensions, the surface of

constant phase form helical spirals with a pitch of mλ, where λ is the vacuum wavelength. Helicity of the phase front means that the wavevectors have azimuthal

2. HISTORY

6

components that circulate around the core22. Owing to this circulation, the optical wave carries orbital angular momentum23. The pioneering work on helical-wavefront waves was reported by Bryngdahl24, 25, while the actual term “optical vortex” was offered by Coullet et al.26. Screw and edge dislocation in optical wavefronts were analyzed in detail by Nye and Berry6, 8, 27, 28 and Wright29. Screw phase dislocations were first observed in laser output beams30 and one of the most familiar examples of OV’s is the TEM*01 “doughnut” mode of a laser cavity26, 31-34

(shown on Fig. 2.1). Multiple vortex modes in various resonators were also

investigated; propagation dynamics, stable spatial structure and creation and annihilation of vortex modes have all been extensively explored35-45. Field vortices were also found in laser speckle patterns, produced by surface scattering of optical waves7, 46. Beams with spiral wavefronts were considered theoretically by Kruglov and Vlasov47 and by Abramochkin and Volostnikov48. Experimental means to obtain optical vortices include active laser systems30, 38, 49, 50

, low-mode optical fibers51, alteration of the beam wavefront with a phase mask10, 52,

computer-generated holography53-57, and a mode-converter58-60. Optical vortices were also found to occur in non-linear optics9, 61-65, where their most attractive feature is the possibility of creation of optical vortex solitons10, 11, 66-69. Propagation of OV’s in selfdefocusing media70, 71, vortex creation via nonlinear instability66, 72, 73 , quantum nonlinear optical solitons74,

75

, polarization effects67, propagation dynamics of vortices76-83, and

vortex arrays84, 85 , as well as other effect86, were investigated. More recently, an interest in optical vortices was sustained by their similarities with quantised vortices in superfluids and Bose-Einstein condensates87-90.

2. HISTORY

7

A single vortex placed within a coherent scalar beam in a paraxial approximation may be expressed by the complex electric field:

E ( r ,φ , z , t ) = A(r ′, z ) g ( r , z ) exp[i ( mφ ′ + β + 2πz / λ − ωt )]

(2.1)

where coordinates ( r , φ ) and primed coordinates ( r ′ , φ ′ ) are measured with respect to the center of the beam and the vortex core, respectively, ω is the angular frequency, t is the time, λ is the wavelength, g ( r , z ) and A( r ′, z ) are respectively the beam envelope and the vortex core profile in the transverse xy plane at z. The field phase β is an arbitrary constant. We will often consider the beam in the initial plane at z = 0, where

g (r ) and A(r ′) will denote beam envelope and vortex core profile in that plane. Perhaps the most typical and mathematically simple description of a beam distribution are a Gaussian envelope: g (r ) = E0 exp(− r 2 w02 ) , and a plane wave: g (r ) = E0 . Here, E0 and w0 are a measure of the field amplitude (assumed to be real) and characteristic beam size, respectively. The more common vortex core profiles belong to “large-core”52,

91-96

,

A( r ′ )=| r ′ / w0 ||m| , and “small-core” (“point” ) vortices57, A( r ′ )= tanh (r ′ / ws ) . Here m m

again is the topological charge and ws is the size parameter, ws 0 )

and below

(x ≈ 0, y < 0 )

the phase singularity, the value of the factor

cos (mφ + β + k x x ) switches sign, thereby creating a forking pattern. If we encircle the vortex core, then the number of fringes entering the circled area differs from the number of exiting fringes by m. If we choose the circle around a positively charged vortex so that the number of entering fringes is 1, then inside the area that fringe splits into m+1 fringes. Conversely, if vortex has a negative charge, then inside the circle m+1 entering fringes will combine into a single fringe. Therefore analysis of interference fringes identifies location and topological charge of a vortex. A related split beam technique may be used when a coherent reference beam is not available31, 99. Other detection methods include the mode converter technique58, 59 and holography93, 100.

2. HISTORY

9

The above description of optical vortices assumes a well defined phase and a strict relation between phases at different points in the beam, i.e. different points in the beam are correlated or coherent. In the real world, however, both intensity and phase of any point of a beam may fluctuate randomly. In partially coherent light, where statistics are necessary to describe the phase, the description of vortices is more complicated. As a model, one may assume that the phase β is varying randomly with time and location, and employ a correlator to describe the mutual coherence (correlation) of pairs of points in the beam.

2.3. Partial coherence The concept of coherence may be traced back to the pioneering experiments by Young in 1807101 and Fresnel in 1814-1816102-104 on the interference of light from two pinholes. The fundamental result of the experiment was the understanding that in order to observe the interference of light, the two pinholes must be illuminated by a single source. If each pinhole is illuminated by an independent source than the phase difference between two sources is generally a random value. The changes in phase difference happen so fast that the typical integration measurement tool (eye, film, or camera sensor) registers a certain average of the instantaneous interference patterns. This may happen when light comes from two different points of the same source or for light coming from a single point but at different instants of time. If no interference is observed then such non-interfering beams are called incoherent. In Young’s original experiment, the conditions of small path difference and smallness of the source were both taken into account to create two beams capable of interfering. The term “coherent” was used to describe interfering beams.

2. HISTORY

10

To quantify the coherence properties the concept of coherence volume may be used. Here we only give a brief account of the necessary definition and refer the reader to the appropriate textbooks 17, 18. Interference fringes in Young’s experiment (see Fig. 2.1) were found to form only if ∆θ ∆s ≤ λ , where ∆θ is the angle that the distance between the pinholes subtends at the source, ∆s is the source size, and λ is the mean wavelength. If R is the distance between the plane containing an incoherent (e. g. thermal) source and the plane containing the pinholes (see Fig. 2.1), than in order to observe the interference fringes, the pinholes must be located within a region around the axial point Q in the pinhole plane. This region is called the coherence area of light in the pinhole plane and its area is on the order of aC ≈ ( R∆θ ) 2 . The square root of the coherence area is lC , sometimes called the transverse coherence length. On the other hand, interference fringes are not observed when the time delay between two interfering beams coming from the same source is greater than the coherence time, τ C , given by the order-of-magnitude relation τ C ≈ 1 / ∆ν

with the ∆ν

being the spectral width. The corresponding

longitudinal coherence length is defined by LC = cτ C ≈ c / ∆ν = λ 2 / ∆λ . The volume of a cylinder with the bottom area equal to the coherence area, and the height equal to the LC , defines the coherence volume. The understanding of partial coherence came into optics as a result of the work of Van Cittert, who calculated the partition functions, showing the correlation between amplitudes in different points105, 106; and Zernike, who actually introduced the notion of degree of coherence and mutual intensity107. Several authors made major contributions to the development of the concept of coherence in the 1950’s, including Hopkins108,

109

,

2. HISTORY

11

Parrent110, and Wolf111, 112. The concepts of both the second-order coherence (correlation between points in the field)113,

114

and the forth-order correlation between photons

(correlation of intensities)115-117 emerged. The development of the coherence theory rapidly continued with the invention of the coherent source (laser) and led to the creation of the quantum theory118-120 for optics, which has now developed into the field of quantum optics121. Probably one of the first summaries of coherence properties of optical fields is the review by L. Mandel and E. Wolf

122

. The coherence theory has developed

further with numerous papers published and several textbooks available17, 18, 123, 124. In this thesis own main interest is the application of coherence theory to optical vortices. The methods of quantitative description of the coherence, the characteristics of the partially coherent light sources, and the propagation equations for the partially coherent beams, will be central in this description.

2.4. Mutual coherence function We are normally concerned with stationary fields. Moreover as a rule the fields are also assumed ergodic, i.e. time and ensemble averages are equivalent. The mathematical description of the second-order coherence theory is based on the mutual coherence function (MCF)18, 112, which may be defined as: r r r r Γ(r1 , r2 ,τ ) =< E1 (r1 , t + τ ) E2* (r2 , t ) >

(2.2)

where < > denotes the averaging over time t, and τ is the time delay between fields E1 r r and E2 at transverse plane points r1 and r2 respectively. The subscript j=1, 2 on the

r electric field Ej emphasizes that the field is a function of r j . The MCF is a

2. HISTORY

12

multidimensional function and may often by characterized by two derivative functions: the [average] intensity:

r r r I (r ) = Γ(r , r ,0)

(2.3)

r r r Χ(r ) = Γ(r ,−r ,0)

(2.4)

and the [average] cross-correlation:

From a physical standpoint, it is convenient to express MCF as a product of the field r r r r E1 (r1 ) , its complex conjugate field E2* (r2 ) , and a correlator C (r1 , r2 ) : r r r r r r Γ(r1 , r2 ) = E1 (r1 ) E2* (r2 ) C (r1 , r2 )

(2.5)

Alternatively, MCF may be expressed in the form:

r r r r r r Γ(r1 , r2 , τ ) = I (r1 )1 / 2 I (r2 )1 / 2 γ (r1 , r2 , τ )

(2.6)

r r r r r r where γ (r1 , r2 ,τ ) = Γ(r1 , r2 ,τ ) /[ I (r1 ) I (r2 )]1 / 2 is the normalized MCF, also known as the complex degree of coherence18. The description of spectral effects of partial coherence125, 126 may be based on the r r cross-spectral density W (r1 , r2 , ω ) (CSD), defined by the equation18:

r r ~ r ~ r < E1 (r1 , ω ) E 2* (r2 , ω ′) >= W (r1 , r2 , ω )δ (ω − ω ′)

(2.7)

~ r ~ r where E1 (r1 , ω ) and E 2 (r2 , ω ′) are electric fields having frequencies ω and ω ′ in the angular frequency domain (the wave sign on top of a field specifies the frequency domain, transformation between time and frequency domains is described by a Fourier integral18), δ (ω − ω ′) is the delta-function.

2. HISTORY

13

2.5. Partially coherent sources One of the cornerstones of coherence theory, and optics in general, is the description of the source of light. Coherence properties of the source may be described with the spatial r r coherence function Γ(r1 , r2 ) , where we consider all points of the source at the same

instant of time and drop τ = 0 from notation. We usually restrict our attention to the 2dimensional planar sources127-129, although 3-dimensional sources have also been investigated130, 131. In recent years, the so-called Schell-model sources132,

133

have been playing an

increasingly important role in optical coherence theory134-140. For a Shell-model source, r r the complex degree of coherence between two points depends on r1 and r2 only through r r r r r r the difference r1 − r2 : γ (r1 , r2 ) = γ (r1 − r2 ) . For modeling purposes in this thesis, we

routinely use sources with the Gaussian-Shell correlator141, where the Schell-model type correlator has a Gaussian distribution: r r r r 2 C (r1 − r2 ) = exp(− r1 − r2 / lC2 )

(2.8)

and lC is the transverse coherence length. Further investigation has led to the development of the Carter-Wolf source128, r r r r r r where MCF is expressed in the form: Γ(r1 , r2 ) = I (r1 + r2 )γ (r1 − r2 ) . Here the intensity is a r r r r function of the sum of coordinates r1 and r2 , and γ is a function of r1 − r2 . The separable

intensity and γ of a Carter-Wolf source greatly simplify the propagation analysis of the MCF. For example, by making transition to the average and difference variables, r r r r r r r = r1 + r2 and r12 = r1 − r2 , the spatial Fourier transform of the MCF becomes a product

2. HISTORY

14

of Fourier transforms of intensity and γ . The limitation of the Carter-Wolf source is the requirement for the intensity profile to be broad compared to that of spatial coherence128.

2.6. MCF investigation, vortex The possibility of obtaining non-diffracting sources137, 142 sparked further interest in the partially coherent sources. Bessel Schell-model sources produced with circular apertures were analyzed using modal expansion143, followed by more thorough study of partially coherent Bessel-Gauss beams with a Gauss-Schell complex degree of coherence144. The next step was the application of modal expansion to partially coherent sources with helical modes145 and investigation of their propagation146. Partially coherent beams with separable phase, carefully constructed to carry an optical vortex modes, have been recently studied analytically147 and experimentally148. Recently, the actual structure of the MCF and CSD has been investigated. For example, the phase singularities in the CSD were analyzed using partially coherent light emerging from two pinholes149. Modal expansion was used to predict vortices in CSD150. Spatial Correlation Vortices (SCV’s) in the cross-correlation function of a partially coherent beam with an off-axis vortex were analyzed numerically151, analytically152 and confirmed experimentally153. In the case of a partially coherent beam with an on-axis vortex, the dislocation ring in the cross-correlation was observed even for low-coherent light154. Thus, the cross-correlation proved to be a more robust description of coherence effects in singular optics than the intensity.

2. HISTORY

15

2.7. Propagation, beam spread The coherence properties of a beam may change upon propagation107,

111, 112

. Most

notably the light, propagated from an incoherent source, becomes partially coherent18. That result is the essence of Van Cittert-Zernike theorem107. A real life example of partial coherence is the image of a mesh fence, composed of thin wires and illuminated by a incoherent source of small, but non-zero angular size (sunlight or a thermal source, placed at a sufficiently large distance from the wires). The light at the fence is partially coherent. The reader may readily verify that although the contrast of an image on a screen decreases as we move the screen away from the fence due to diffraction155, the wires become clearly visible when the screen is placed close to a fence at a distance much greater than the wire thickness. In comparison, wires are not visible when an incoherent light source (f. e. a flashlight covered with a diffuser) is placed directly behind the fence. The effect of wire visibility is most easily understood by considering the angular beam spread156-159, which describes how the major portion (fraction ε=0.96) of the beam power is spread as the distance from the source is increased17. For a Gaussian CarterWolf r

source, r

characterized r

by

r r r r I (r1 + r2 ) = I G exp(− | r1 + r2 |2 / 2w02 )

and

r

γ (r1 − r2 ) = exp(− | r1 − r2 |2 / 2lC2 ) , the radius of a circle that encloses ~96% of the beam power is given by the equation RCW = [2πw02 (1 + z 2 / k 2 w02lC2 )]1 / 2

(2.9)

2. HISTORY

16

where z is the distance from the source, w0 is the Gaussian beam size at the source, and k is the propagation constant (modulus of the wave-vector) 17. The assumption w0 >> lC is used. In the limit z >> kw0 lC (far from the source), the ratio of RCW to z is inversely proportional to the coherence length: RCW / z ≅ (2π )1 / 2 / klC

(2.10)

On the other hand, next to the incoherent source the light goes in all directions and RCW / z ≅ (2π )1 / 2 w0 / z , i.e. for small value of z the ratio of RCW to z may be infinitely large (incoherent beam). The mathematical description of the propagation of partially coherent beams is largely based on the Rayleigh-Sommerfeld diffraction theory160. The Rayleigh diffraction formula of the first kind, derived in 1897, provides the solution to the Dirichlet boundary problem: r r r ∂ E (r , z ) = −1 / 2π ∫∫ E (r ′,0) [exp(ikR) / R]dr ′ ∂z z =0

(2.11)

r r r where E (r , z ) is the field at a point r in the transverse plane z ≥ 0 and E (r ′,0) is the

r field at a point r ′ in the transverse plane z=0, k is the wave number, and R is the distance r r between points (r ′,0) and (r , z ) (see Fig 2.2). At the limit R >> λ ( 1 / R > σ cr | x s | ≅ 1 + 2exp( −4 σ2 x s)

(4.15)

These analytical results and the numerically calculated intensity profiles in Fig. 4.2 demonstrate that the vortex core may be readily repositioned by changing the relative position or phase of the component beams. From an experimental and application point of view the later variation is often easier to achieve in a controlled linear fashion. For example, the net field may be constructed from two collinear beams from a MachZehnder interferometer, whereby the phase is varied by introducing an optical delay in one arm of the interferometer. A demonstration of the phase sensitive vortex trajectory is shown in Fig. 4.3 for the case σ=0.47 (i.e., σ0 plane, then k1z = (Κ 2 − k12x − k12y )1 / 2 and

5. SPATIAL CORRELATION VORTICES: THEORY

80

k 2 z = (Κ 2 − k 22x − k 22y )1 / 2 , where K is the wavenumber, Κ = 2π / λ , same for both

wavevectors. Earlier Carter and Wolf obtained expression for the k-space MCF for m=0 by analytical integration of Eq. (5.11)131: r r rr r r rr Γm∞=0 (k1 , k 2 , z = 0) = E02 [π 2 (a 2 − b 2 )] exp(−αk1k1 − αk 2 k 2 + 2 β k1k 2 )

(5.16)

where α = a /[4(a 2 − b 2 )] and β = b /[4(a 2 − b 2 )] . To describe the more general case (m ≠ 0) we make use of the relation: r r ∂ p + q +u + v Γm∞ (k1 , k 2 , z ) i = ∂k1px ∂k 2qx ∂k1uy ∂k 2v y r r r r r r r r p q u v [ x x y y Γ ( r , r , z ) ] exp[ − i ( k 1 2 1 2 m 1 2 1 ⋅ r1 − k 2 ⋅ r2 )]dr1 dr2 ∫∫ p +u − q −v

(5.17)

where p, q, u, v are positive integers. Eq. (5.17) is obtained by taking partial derivatives of the left and right parts of Eq. (5.11). Eq. (5.17) may govern the calculations of the kspace form for any MCF, which can be expressed as a product of a polynomial and an MCF of a Gaussian field with a Gaussian-Shell correlator. In the case of a LaguerreGaussian vortex of charge m=1 we obtain: r r ∂2 ∂2 ∂ ∂ 1 Γm∞=1 (k1 , k 2 , z ) = 2 [ + + s 2 − is ( − )+ ∂k1x ∂k 2 x w0 ∂k1x ∂k 2 x ∂k1 y ∂k 2 y r r ∂ ∂ ∂ ∂ ∂ ∂ s( + ) + i( − )]Γm∞=0 (k1 , k 2 , z ) ∂k1 y ∂k 2 y ∂k 2 x ∂k1 y ∂k1x ∂k 2 y

(5.18)

Calculation of the derivatives and multiplication by the transfer function T (k1z , k 2 z ) result in:

5. SPATIAL CORRELATION VORTICES: THEORY

81

r r r r 1 Γm∞=1 (k1 , k 2 , z ) = 2 {4[ β + (α 2 + β 2 )k1 ⋅ k 2 − αβ (k12 + k 22 ) w0 + i (α 2 − β 2 )(k 2 x k1 y − k1x k 2 y )] + 2 s[( β − α )(k1 y + k 2 y ) − i (α + β )(k 2 x − k1x )] + s 2 } r r × exp[i ((Κ 2 − k12x − k12y )1 / 2 − (Κ 2 − k 22x − k 22y )1 / 2 ) z ]Γm∞=0 (k1 , k 2 ,0)

(5.19)

where we assume propagation into z>0 plane. r r r r Substitution of k1 = k and k 2 = − k ( k1z = k 2 z ) gives us the cross-correlation function151:

r r r r 1 Γm∞=1 (k ,− k , z ) = 2 {4 β + 4(α − β ) 2 (k x2 + k y2 ) + 4is (α + β )k x + s 2 }Γ0∞ (k1 , k 2 ,0) w0

(5.20)

r where ( k x , k y ) are k coordinates. We now substitute the beam size w0 and the transverse

coherence length lC into coefficients α and β: r r r r Γm∞=1 (k ,− k , z ) = ( w02 / 4){− k 2 + 4isk x / w02 + kV2 }Γm∞=0 (k ,− k ,0)

(5.21)

where kV = 2(1 / σ + s 2 / w04 )1 / 2

(5.22)

and σ = lC2 + 2w02 is the beam spread. We note the right part of Eq. (5.21) is independent of z and therefore the cross-correlation function in the k-space is invariant upon propagation. Further analysis of Eq. (5.21) for s = 0 confirms the presence of a characteristic ring154 of radius k R = 2 / σ 1 / 2 . In the case s ≠ 0 the cross-correlation contains two zeros151, 188, located at points (0, ± kV ). One may verify that the cross-correlation is singular at these points by rewriting polynomial part of Eq. (5.21) in terms of new coordinates (k x′ , k ′y ) = (kx, ky ± kV ): r r r r Γm∞=1 (k ,− k , z ) = {−( w02 / 4)(k x′ 2 + k ′y2 ) + isk x′ ± k ′y kV }Γm∞=0 (k ,− k ,0)

(5.23)

5. SPATIAL CORRELATION VORTICES: THEORY

82

In the vicinity of zeros the polynomial term becomes isk x′ ± k ′y kV , defining vortices of charge m = +1 at (kx, ky)=(0, kV ) and m = −1 at (kx, ky)=(0, − kV ). Fig. 5.7 shows the unit-less distance d = 2kV w0 between zeros in the crosscorrelation as a function of s / w0 and lC / w0 , where kV is given by Eq. (5.22) (for s=0 the unit-less diameter of the characteristic ring 2k R w0 is shown instead). Analysis of Eq. (5.22) reveals that in the coherent limit ( lC → ∞ ) σ → 0 and kV is a linear function of vortex displacement s: kV = 2s / w02

(5.24)

In the incoherent limit ( lC → 0 ) the beam spread is defined by the beam size, σ → 2w02 , and we obtain: kV = (4s 2 + 2w02 )1 / 2 / w02

(5.25)

In the limit s } cos(k[c x x + c y y ])

(6.1)

where ∆ is the length difference between interferometer arms, and c x , c y are constants defined by the interferometer alignment. For a slowly varying field distribution we may define Imax(x, y) and Imin(x, y) as local maximum and minimum intensities of the fringe pattern. The fringe visibility, defined as

η ( x, y ) = ( I max ( x, y ) − I min ( x, y )) /( I max ( x, y ) + I min ( x, y )) , is related to the cross-correlation Χ ( x, y ) as:

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

η ( x, y ) = 2 Re{Χ( x,− y ) exp(ik∆)} /{I (− x, y ) + I ( x,− y )}

93 (6.2)

Assuming I (− x, y ) ≅ I ( x,− y ) , real and imaginary parts of the normalized crossr correlation χ (r ) are given by the visibility η ( x, y ) for ∆ = 0, λ/4 respectively. The phase r features of χ (r ) may be determined from the fringe deformation.

6.4 Experimental Observation of Spatial Correlation Vortices As we explained earlier, a vortex may be created by transmitting a beam with a planar phase front through the phase mask of variable thickness. A schematic diagram of partially coherent light transmitted through a vortex phase mask is shown in Fig. 6.1. A vortex mask of diameter 2w is placed in the plane z=0 at a distance zs from an extended incoherent source of diameter Ds. The transverse coherence length at the mask is approximately given by lC = 1.28λz s / Ds , where λ is the wavelength of a source. The MCF in the detection plane, located at a distance zd >>λ from the phase mask (see Fig. 6.2), for a narrow-band source and a small-angle approximation18 is approximately given by18: r r r r r r Γ(rd 1 ,rd 2 , z d ) ≈ ( z d / λ ) 2 ∫∫ Γ(r1 , r2 ,0)exp[ik ( R1 − R2 )]R1−2 R2−2 dr1dr2

(6.3)

z =0

r r r r where Γ(r1 , r2 ,0) may be described by Eq. (5.5), rd 1 and rd 2 are two points in the

r r 2 detection plane, Ri2 = ri − rdi + zd2 (i=1,2), and k is the wave number, k= 2π / λ .

Our experimental setup is shown in Fig. 6.3. Broadband light from an incoherent halogen source passes through two apertures. The spatial coherence is controlled by varying the aperture Ap1 radial size A (0.25mm to 1.5mm with an increment of 0.125mm

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

94

and precision of ± 5nm), while the second aperture Ap2 has a fixed radial size w = 2.5mm. The distance between the apertures is B = 42cm. The transverse coherence width lc at the second aperture is approximately given by lC = 0.64(λ z s / A) 17, where λ is the

average wavelength. Aperture Ap2 is imaged with unity magnification by the lens system L1 of focal length f1 = 25mm into a vortex phase mask (PM) plane z=0, designed to

produce an m = 1 vortex at wavelength λ = 890nm and air interface. Phase mask transverse position in the plane is controlled with a micrometer, allowing on- and off-axis center location with a precision of 1nm. The resulting beam is passed through the WFFI, which is built from a standard Mach-Zehnder interferometer by incorporating two Dove prisms DP1 and DP2 to fold the wavefront in each arm, one about the horizontal, and second about the vertical axis. The interferometer is aligned for zero path difference between the arms to allow for the white light interference and investigation of Re χ ( xd ,− yd ) in the detection plane z = 92mm. The detection plane is imaged with a magnification of 3 by two lenses having focal lengths, f2 = 125mm (L2 ) and f3 = 500mm (L3 ) onto a cooled CCD camera (Mead model 416XT). A 50nm band pass filter with a mean transmitted wavelength of λ =800nm is attached to the front of the camera to achieve quasi-monochromaticity and minimize temporal coherence effects. Observed interference fringes allow for the forking analysis4,

7

of the cross-

correlation. The fringe visibility is appreciably different from zero inside the coherence area17, which border (ring of zero fringe visibility) may be defined by the Airy disk190. The coherence area location and diameter are verified by removing the phase mask from a beam. We analyze the fringe structure inside the Airy disk as a function of aperture Ap1

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

95

size A and off-axis displacement s of the vortex phase mask. Fig. 6.4 illustrates the experimentally obtained distribution of intensity, defined by Eq. (6.1) for ∆ = 0, for the partially coherent light source with a charge +1 embedded vortex, a1 = 0.5mm (see Fig. 6.4(a),(b)) and a1 = 1mm (see Fig. 6.4(c),(d)). For an on-axis vortex (s = 0, see Fig. 6.4(a),(c)) we observe a dislocation ring, not present for a plain-wave beam. We note that zero fringe visibility at the ring indicates zero of the real part of MCF. As we move the vortex off-axis (displacement s > 0), the dislocation ring breaks into two oppositely charged composite spatial correlation vortices (see Fig. 6.4 (b),(d)), identified by the vertex of fringe forks. We note that the distance between SCV’s is larger for the more coherent beam (see Fig. 6.4 (b)) illustrating SCV’s dependence on the coherence length of a beam. One may notice that due to the wavelength mismatch between our band pass filter ( λ = 800nm ) and the phase mask (λ = 890nm) additional vortices may form inside the coherence area (see Fig. 6.4(a), quadrants 1&3 ). That effect is not related to the coherence properties and may be attributed to fractional vortices164. Fig. 6.5 shows the displacement of SCV’s from the center of symmetry as a function of s (if dislocation ring is formed instead, the ring radius is shown). Experimental data (marks) are compared with numerical results, obtained in151(curves). Numerical data were calculated in the far field for a Laguerre-Gaussian vortex151 and converted to metric terms using experimental values. Since in Chapter 5 we were only able to calculate vortex separation for s/w0 < 0.6, here we used 4-th order polynomial interpolation to calculate d for larger s/w0. The behavior of SCV’s in our experiments and

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

96

simulations are qualitatively similar for s/w0< 0.5. The growing difference at large values of s may be attributed to our use of Laguerre-Gaussian approximation in numerical calculations and, more fundamentally, to the use of Gaussian-Schell correlator. While the Gaussian-Schell model is robust for low values of s, for large separation values the presence of an Airy disk in the experimentally obtained cross-correlation indicates that the actual coherence properties at vortex locations may differ significantly from the model. Correct analysis may require Bessel-type correlator. We also note that the experimental and numerical data are obtained in different transverse planes. However as we have shown in Ch. 5, SCV’s separation in k-space may be expected to remain invariant upon propagation, and thus in x-space it may be described by Eq. (2.4) for the beam spread. The main qualitative difference between two planes is expected to be the vortex rotation due to Gouy phase shift (for the coherent case see 4, p.117). For a completely coherent beam Fig. 6.5 indicates that the vortex dislocation ring size is zero (spatial singularity annihilate), while SCV’s turn into composite vortices, described in Ch. 4. As the coherence length decreases, the dislocation ring size for an onaxis vortex increases154. This may be understood if we recall (Ch. 5) that real part of r

χ (r ) for a vortex beam of topological charge m=1 is negative151. The vortex presence may be associated with the anti-correlated (negative real part) cross-correlation. As the coherence decreases, the area of positive correlation in the center of cross-correlation grows in size and negative part moves out of the field of view, resulting in growing ring size / vortex separation, while the rate of motion of SCV’s [ ∂d / ∂s ] decreases.

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

97

6.5 Geometrical Model Interpretation Fig. 6.1 provides a geometrical description of partially coherent light propagating through an off-axis vortex phase mask. The transverse plane coordinates of a center of a mask are (s, 0). In the detection plane z=zd a conical projection of the source through the center of a mask forms a circular region of radius rd = Ds ( z d / z s ) with the center at point ( s d , 0), where Ds is the size of a source, zs is the distance between source and mask planes, and

sd = s (1 + zd / z s ) . Light collected inside this region contains rays from all sectors of the mask, while an outside observer doesn’t see the mask center and collects rays from only adjacent sectors of the mask. Cross-correlation in the detection plane may be illustrated using Fig. 6.2. The light at points ( xd , yd ) and ( − xd ,− yd ) may be represented by projections of the source ‘Beam1’ and ‘Beam2’, the second projection is through an “imaginary mask”, located at the point (–s, 0) in plane z = 0. In the coherent limit ( Ds → ∞ ) rd → 0 and the crosscorrelation contains singularities at points ( ± sd , 0). As the coherence decreases, rd grows and now points ( ± sd , 0) are filled with light from all segments of a corresponding phase mask. Careful calculations (see Ch. 5 and 151) show that SCV’s are displaced in the horizontal direction to a new location further away from each other.

6.6 Conclusion Spatial correlation vortices were detected in the mutual coherence function using a wavefront folding interferometer and forking analysis. SCV’s behavior was analyzed and

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

98

found to depend on the coherence length. We showed that SCV’s response to the changes in off-axis displacement of original vortex is determined by the coherence length and varies as the SCV’s move outside the central area of the cross-correlation function.

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

Fig. 6.1:

99

A diagram of partially coherent light transmitted through a phase mask. Light from an incoherent light source of transverse size Ds in the plane

z = z s propagates to the phase mask plane z=0. The center of the mask is displaced from the optical axis in horizontal direction by distance s. In the plane z = z d a conical projection of the source through the center of the mask forms an enclosed circular region.

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

Fig. 6.2:

Cross-correlation is the correlation between points ( xd , yd ) and

( − xd ,− yd ) , and may be described by interfering two symmetrical projections ‘Beam1’ and ‘Beam2’.

100

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

Fig. 6.3:

101

An experimental setup shows light from a spatially incoherent light source (halogen bulb) with a mean wavelength of 800nm and a bandwidth of 50nm passing through two apertures separated by a distance B = 42cm. The Ap2 plane is imaged with unity magnification by a lens system L1 (focal length f1 = 25.1mm) onto a phase mask PM, designed to produce an m=1 vortex at wavelength λ=890nm and air interface. Phase mask

transverse location is controlled with a micrometer, allowing on- and offaxis mask center location. The plane z=92mm is imaged with a magnification of 3 by two lenses having focal lengths, f2 = 125mm (L2 ) and f3 = 500mm (L3 ) onto a cooled CCD camera (Mead model 416XT) after passing through a wavefront folding interferometer comprising Dove prisms, DP1 and DP2.

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

Fig. 6.4:

102

Experimental interferograms of the cross-correlation function of a vortex beam of charge m=1. Partially coherent light at the phase mask has a radial size w=2.5mm, and relative coherence length lc/w of 0.34 (a-b) or 0.17 (df). The imaged plane is located at a distance of 92mm from the phase mask. The phase mask off-axis displacement s is: (a, c) 0µm, on-axis vortex, dislocation ring is observed; (b, d) 7µm, off-axis vortices in the fringe pattern are present.

6. SPATIAL CORRELATION VORTICES: EXPERIMENT

Fig. 6.5:

103

Distance between spatial correlation vortices d vs. the off-axis displacement s for coherent and partially coherent light for partially coherent and highly coherent (lc/w=0.17; 0.34; 340) beams. Markers show experimental points, curves – results of numerical modeling.

1-20 6-11, 13, 14, 17, 18, 21-183 22, 76, 151, 184-188 189, 190

,

.

7. SPONTANEOUS DOWN CONVERSION

7.1 Introduction In previous chapters we considered a partially coherent beam with a vortex embedded into it, i.e. an already partially coherent beam was sent through a vortex phase mask. In this last chapter we consider an example of a process where a coherent vortex beam is sent through an optical system, which then affects the beam’s statistical properties. In recent years the investigation of parametric scattering regained the interest, in particular due to the question of what happens to an optical vortex and associated orbital angular momentum of light191 in a 3-photon processes. Here we consider the degenerate down-conversion, where a pump photon of frequency ωpump is split into signal and idler photons of the same frequency ωsignal = ωidler. In Sec. 2 we shall briefly discuss the orbital angular momentum of photons in a vortex beam. Sec. 3 will explain the phase-matching conditions, which govern the process of down-conversion. In Sec. 4 we shall present quantum and wave models, which may be used to find spatial distributions of signal and idler beams. Numerical results are presented in Sec. 5, followed by conclusion.

7. SPONTANEOUS DOWN CONVERSION

105

7.2 Optical Vortex and Orbital Angular Momentum The vortex phase factor exp(imφ ) , where m is the topological charge (m = 0,

± 1, ± 2,

…), and the resulting helical phase front of beam are responsible for the orbital angular momentum (OAM), carried by photons in a vortex beam. For example, each photon in a Laguerre - Gaussian (LG) mode beam, described in the transverse plane ( r ,φ ) by:

Em ( r , φ ) = E0 | r / w0 || m | exp( − r 2 / w02 ) exp( imφ )

(7.1)

carries m quanta of OAM. Here we follow notation of Ch. 2, where w0 is defined as the size of a Gaussian beam. In classical depiction, schematically shown on Fig. 7.1 the OAM of photons in a vortex beam may be understood by expressing the transverse k-vector component of a beam as a gradient of the phase: r r k ⊥ = −∇φ = − mφˆ / r

(7.2)

where φˆ is the azimuthal unit vector. We may then express the mechanical angular momentum of a photon as: r r r L = hr × k ⊥ = hmzˆ were zˆ is the axial unit vector (normally the direction of beam propagation).

(7.3)

7. SPONTANEOUS DOWN CONVERSION

106

7.3 Phase-Matching Conditions A 3-photon process of spontaneous down-conversion is a conversion of a pump photon into signal and idler photons as a result of a non-linear interaction with a crystal (see diagram on Fig. 7.2). The so-called phase-matching conditions, recognized as the conservation laws for the momentum and energy, may be used to govern the process of conversion. We define the wave-vector mismatch and the frequency mismatch as r r r r r r r ∆k = k p − k s − ki and ∆ω = ω p − ωs − ωi respectively, where k p , k s , ki are k-vectors for

the pump, signal, and idler waves, and ω p , ωs , ωi are the corresponding frequencies. The momentum conservation results in the wave-vector matching condition: r ∆k = 0

(7.4)

and energy conservation hω p = hωs + hωi results in the frequency-matching condition: ∆ω = 0

(7.5)

r r Together Eq. (7.4) and (7.5) describe the conservation of the wave phase term ( k ⋅ r − ωt )

in a sense that the phase of a pump wave is equal to the sum of phases of signal and idler waves. We limit ourselves to the process of degenerate parametric scattering, i.e. | ωs | = | ωi | , and start with an assumption that the OAM is also conserved. Conservation of OAM may be expressed as the topological charge conservation condition, which states that the topological charge mp of a pump beam equals to the sum of topological charges of signal (ms) and idler (mi) beams. In other words, in the process of conversion the topological charge mismatch ∆m = m p − ms − mi is expected to be zero:

7. SPONTANEOUS DOWN CONVERSION ∆m = 0

107 (7.6)

We may define the total phase of a beam as: r r Φ = k ⋅ r − ωt + mφ

(7.7)

In the case when all three phase-matching conditions are satisfied, the total phase of a pump wave is equal to the sum of total phases of signal and idler waves: r r ∆Φ = Φ p − Φ s − Φ i = ∆k ⋅ r − ∆ωt + ∆mφ = 0

(7.8)

where Φ p , Φ s , Φ i are phases of pump, signal, and idler waves respectively. Immediate question that arises is what happens to a vortex pump beam of topological charge mp=1. Given that topological charge of a vortex beam is an integer quantity, the topological charge is expected to be distributed between signal and idler waves in an asymmetric fashion. Physically the quantum of OAM from the pump photon may be expected to go randomly to the signal or idler photons. We note two different types of phase-matching available for investigation. In both cases the pump wave has to be extraordinary polarized. In a type-I process both signal and idler waves have extraordinary polarization: e p = es + ei , while in a type-II process the signal wave is extraordinary polarized and the idler wave has ordinary polarization: e p = es + oi , where e and o denote an extraordinary and ordinary waves. The major advantage of the type-II phase-matching is the possibility to distinguish photons in signal and idler beams using a simple polarizer. When desired, signal and idler beam can also be made indistinguishable by using a 450 polarizer. For that reason in our experiments and simulations we use type-II phase-matching.

7. SPONTANEOUS DOWN CONVERSION

108

7.4 Signal spectrum We assume that the amplitude of a pump field inside the crystal is described by the Laguerre-Gaussian mode profile, given by Eq. (7.1). We also assume the non-focusing regime and non-depletion by the conversion process, i. e. constant pump beam transverse profile along the axial direction z. That assumption is valid in a case of thin crystals and low conversion efficiency. A down-conversion signal may be described using the angular (k-space) distribution. In a general case the spatial-temporal spectrum of signal photons is given by121: r r 2 r g (ωs , k s ) = C1 ∫ δ (∆ω ) f (∆k ) dki

(7.9)

V

where integration is conducted via the idler k-space volume V of a non-linear crystal, r

δ (∆ω ) is a delta-function and f (∆k ) is the conversion form-factor of the crystal. The r shape of f (∆k ) is determined by the crystal properties, as well as by the pump field. In a

simplest case, when microscopic properties of a crystal are defined by the constant r macroscopic susceptibility χ, f (∆k ) may be obtained from Fourier-transformation of the

pump field inside the crystal over the k-vector mismatch: ∞ r r r r r f (∆k ) = ∫ exp(i∆k ⋅ r ) E p (r )dr

(7.10)

−∞

Integration in Eq. (7.9), which describes the photon spectrum, is conducted over r the photon form-factor f (∆k ) 2 , and any information about the phase information of a r pump wave is removed by taking the modulus of f (∆k ) . Hence the photon spectrum has the dimension of intensity. Experimentally that type of spectrum may be obtained by

7. SPONTANEOUS DOWN CONVERSION

109

photon-counting setups. In contrast, the field distribution may be obtained by directly r integrating over f (∆k ) : r r r h(ωs , k s ) = C1 ∫ δ (∆ω ) f (∆k )dki

(7.11)

V

We note the important difference between intensities, calculated using Eq.’s (7.9) and (7.11). The first one provides the number of photons registered by the detector, whereas the second provides the field. Therefore the intensity, calculated from Eq. (7.11) by taking squared modulus, allows taking into account interference effects. If the process of parametric scattering is completely random (such as when the molecules are moving chaotically) and down-conversion signal waves, produced by different combinations of kvectors, do not interfere, then both models may be expected to produce similar results. However in a non-linear crystal one may expect a degree of correlation between various above-mentioned waves and thus the use of quantum theory of optical fields192 may be necessary. In this work we attempt to use the model, described by Eq. (7.11), to consider the parametric scattering of beams with helical phase fronts. r Radial distributions of f (∆k ) , calculated using Eq. (7.10) for Laguerre-Gaussian modes m = 0 (Gaussian beam) and m = 1 (vortex), and normalized to f (0) of a Gaussian beam, are shown in Fig. 7.3. The distributions are plotted as a function of the transverse r radial component ∆k ⊥ , ρ of wave-vector mismatch. As mentioned earlier, we assume constant conversion efficiency throughout a non-linear crystal for a given pump field r r amplitude, k and wavelength. We note that for m = 1 the from-factor f (∆k ) has a minimum at the origin: f (0) = 0 . Therefore in a vortex case the conversion efficiency at

7. SPONTANEOUS DOWN CONVERSION

110

r r ∆k = 0 is zero and a conversion is only possible due to the ‘wings’ of f (∆k ) . In the following calculations we assume a monochromatic detection system, which only detects waves at half the pump wave frequencies. In that case the frequency phase-matching condition is automatically satisfied.

7.5 Quantum and Wave Models For convenience we refer to the photon-counting model, described by Eq. (7.9), as

quantum picture, as opposed to wave picture, described by Eq. (7.11). Signal beam intensity distribution, obtained using quantum picture at ωs = ω p / 2 , has the form: r I s (k s ) =

∫∫

r 2 r f (∆k ) dki

(7.12)

ki − space

In contrast the wave picture gives us the field: ~ r Es ( k s ) =

r r f ( ∆ k )dki ∫∫

(7.13)

ki − space

~ r ~ r and intensity is given by I s (k s ) =| Es (k s ) |2 . Both models may take advantage of the

slowly-varied amplitude approximation:

r f ( ∆ k ) = f xy ( ∆ k x , ∆ k y ) f z ( ∆ k z )

(7.14)

r where transverse component of f ( ∆ k ) is given by: f xy (∆k x , ∆k y ) =

∫∫ E

p

( x, y ) exp i (∆k x x + ∆k y y )dxdy

(7.15)

xy − plane

and longitudinal component for a field, which has a constant transverse distribution within the limits of a crystal, is:

7. SPONTANEOUS DOWN CONVERSION

111

L

f z (∆k z ) = F .T .(1) = ∫ ei∆k z z dz = i(1 − ei∆k z L ) / ∆k z

(7.16)

0

For a Gaussian pump beam the envelope E p ( x, y ) is given by Eq. (7.1) for m=0,

E p ( x, y ) = Em = 0 (r ,φ ) = E0 exp( − r 2 / w02 ) . The integration of Eq. (7.15) results in (see Appendix A): f xy , m = 0 ( ∆k x , ∆k y ) = f xy , m = 0 ( ∆k ⊥ ,θ ) = Akg exp[ − (π∆k ⊥ w0 ) 2 ]

(7.17)

for a vortex pump field the envelope is given by Eq. (7.1) for m=1,

E p ( x, y ) = Em =1 ( r , φ ) = E0 | r / w0 | exp( − r 2 / w02 ) exp( iφ ) , and Fourier transformation results in:

f xy ,m=1 (∆k x , ∆k y ) = f xy (∆k⊥ ,θ ) = Akv ∆k⊥ exp[−(π∆k⊥ w0 ) 2 ] exp(iθ ) where

Ag , Av , Akg , Akv

are

normalization

constants,

and

(7.18)

∆k ⊥ = ∆k x2 + ∆k y2 ,

θ = arctan (∆k y / ∆k x ) are transverse coordinates in ∆k space.

7.6 Numerical Investigation Numerical calculations of the down-conversion signal and idler distributions require the knowledge of properties of the non-linear crystal, which is used for conversion. We modeled parametric scattering in BBO crystal with type-II phase-matching, which was also used in our experimental efforts178. The following constants were used: pump wavelength λp = 364nm, signal and idler wavelengths λs = λi =728nm, beam size w0 = 1mm, crystal length lc = 7mm and the incidence angle (angle between the beam and the crystal optical axis) θ = 480. Calculations followed Eq. (7.9-7.18) and were performed on

7. SPONTANEOUS DOWN CONVERSION

112

Alfa Unix station (21164 processor, 533mhz), numerical grid of 20000x20000 points. Both the signal and idler wave intensity distributions in k-space formed a ring. Part of the signal wave ring, calculated using wave-picture model, is shown on Fig. 7.4 for Gaussian and for vortex (m = 1) pump beam envelopes. Signal distributions, calculated using quantum-picture model, are similar in appearance. For comparison purposes, the experimentally obtained distribution for a vortex pump beam178 is shown on Fig. 7.5. The experimental parameters were the same as the above-mentioned numerical simulation constants, with the exception of spectral distribution, which was defined by 20nm bandpass interference filter, placed in front of the registering digital camera. On Fig. 7.5 both signal and idler rings are shown. We note that signal and idler waves in type-II phasematching have orthogonal polarizations and may be easily distinguished by placing a linear polarizer in a beam. Experimental results were found to be in qualitative agreement with both numerical models. The difference between results, numerically calculated with two models, appear to be in the down-conversion ring profiles. Cross-sections of intensity distributions for a Gaussian and vortex pump beam profiles, calculated with a quantum-model, are shown on Fig. 7.6. We note absence of the appreciable difference between distributions. Therefore one could claim that the vorticity and OAM are destroyed in the process of down-conversion. Early experiments with down-conversion of vortices, where downconversion rings, similar to those shown on Fig. 7.6, were obtained and analyzed, resulted in just such claims193. As we noted earlier, the phase information, contained in kspace distribution of the pump wave, is indeed removed in the quantum-picture model, and therefore that result should not come as a surprise. However, the wave-model

7. SPONTANEOUS DOWN CONVERSION

113

profiles, shown on Fig. 7.7, indicate that the ring profiles for Gaussian and vortex pump beam may be expected to differ. In the Gaussian case our wave-model produced a sincfunction-like profile distribution, in agreement with earlier theoretical results121 and our quantum-model. However in the case of a vortex pump beam, the signal ring profile does not have characteristic zeros of a sinc-function (see Fig. 7.7). The difference in downconversion ring profiles for different pump beam envelopes may indicate conservation of OAM in a beam. The down-conversion ring radius varies with the wavelength of signal wave and therefore experimental observation of exact ring profile would require a bandwidth, approaching a δ-function distribution, as well as a good resolution of the ring structure. In the experimental distribution, shown on Fig. 7.5, neither of these requirements could be satisfied. However experimental confirmation of the conservation of OAM in downconversion of vortex beam was recently demonstrated by Mair at al.179, using photoncorrelation counting technique, and a capability of single photon sorting using interferometers is available191.

7.7 Conclusion We numerically investigated the process of spontaneous down-conversion of optical vortex beams. We derived new phase-matching conditions, which account for the optical vortex OAM. Our modeling, based on new phase-matching conditions, suggested that the vortex in a pump beam might affect the transverse profile of down-conversion rings. Our

7. SPONTANEOUS DOWN CONVERSION

114

results indicated that earlier claims of OAM destruction in the process of downconversion are unsubstantiated.

7. SPONTANEOUS DOWN CONVERSION

Fig. 7.1:

115

Illustration to the notation in Sec. 7.2 . Transverse k-vector component of a beam is a gradient of the phase φ .

7. SPONTANEOUS DOWN CONVERSION

Fig. 7.2:

116

Spontaneous down-conversion in a non-linear crystal is a process of spontaneous conversion of a pump photon into signal and idler photons.

7. SPONTANEOUS DOWN CONVERSION

117

r f (∆k 1.2 )

Gaussian pump

1

0.8

Vortex pump

0.6

0.4

0.2

0 -2.5

Fig. 7.3:

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

r ∆k⊥ , ρ , m −1

Profile of the conversion form-factor for a Gaussian and vortex (m = 1) r pump beams as function of the transverse radial component ∆k⊥ , ρ of

wave-vector mismatch.

7. SPONTANEOUS DOWN CONVERSION

(a)

ks,x

Gaussian pump (m=0)

Fig. 7.4:

(b)

118

ks,x

Vortex pump (m=1)

Part of the signal down-conversion ring, numerically calculated using wave-picture. (a) Gaussian pump (m = 0); (b) Laguerre-Gaussian vortex pump beam (m = 1).

7. SPONTANEOUS DOWN CONVERSION

119

Idler photon

Signal photon

Fig. 7.5:

Experimentally obtained spontaneous down-conversion signal and idler rings.

7. SPONTANEOUS DOWN CONVERSION

120

I/Imax

1.

Quantum picture

.8

Vortex pump (m=1)

.6

Gaussian pump (m=0)

.4 .2 0 -5.00

Fig. 7.6:

-3.33

1.66

0

1.66

3.33

r 5.00−1 k s ,⊥ , ρ , m

Down-conversion signal ring intensity as a function of the transverse radial k-vector component, quantum picture. Intensity is shown relative to the maximum intensity of a Gaussian beam. Numerical data points are fitted with smooth curve profiles.

7. SPONTANEOUS DOWN CONVERSION

121

I/Imax 1.

Wave picture

.8

Gaussian pump (m=0)

.6

Vortex pump (m=1)

.4 .2 0 -5.41 0

Fig. 7.7:

10

–3.74 20

30

-2.08 40

50

-0.42 0.42 60 70

Same as Figure 7.6, wave picture

80

2.08 90

100

3.74 110

r120 130 −1 ks , ⊥, ρ , m

APPENDIX

1-20 6-11, 13, 14, 17, 18, 21-183 22, 76, 151, 184-188 52, 90, 93, 180-183, 189, 190 191-193

,

A. FOURIER TRANSFORM OF LAGUERRE-GAUSSIAN BEAMS

This Appendix contains mathematical derivation of Fourier-space representations of Laguerre-Gaussian beams. The main result of this derivation as Fourier transform for a truncated beam and therefore the main benefactor of this appendix is Ch. 3.

1. Fourier transform of a Laguerre-Gaussian beam

A Laguerre-Gaussian beam in a transverse plane may be expressed is:

Em ( r ,θ ) = E0 | r / w0 || m| exp( − r 2 / w02 ) exp(imθ )

(A.1)

where ( r , θ ) is a set of polar coordinates. In terms of Cartesian coordinates (x, y):

Em ( x, y ) = E0 ([ x + iy ] / w0 )| m| exp( −[ x 2 + y 2 ] / w02 )

(A.2)

The far-field (k-space) distribution of Em ( x, y ) is given by a Fourier transform:

E ∞ (k x , k y ) =



1 w02 −∫∞



∫ E ( x, y) exp(−ik x − ik x

y

y )dxdy

(A.3)

−∞

where (kx, ky) are Cartesian coordinates in k-space. For simplicity we may express farfield distributions of Laguerre-Gaussian fields in terms of dimension-less coordinates tx=(x /w0), ty=(y/ w0) and px=(kx w0), py=(ky w0). In terms of (tx, ty) and (px, py) Eq. (A.3) becomes:

A.FOURIER TRANSFORM OF LAGUERRE-GAUSSIAN BEAMS 124 ∞



E ( px , p y ) =



∫ ∫ E (t , t x

y

) exp(−ip xt x − ip y t y )dt x dt y

(A.4)

−∞ −∞

For a Gaussian field (m=0) Eq. (A.2) in (tx, ty) coordinates becomes:

Em = 0 (t x , t y ) = E0 exp( −[t x2 + t y2 ])

(A.5)

The Fourier transform of Eq. (A.5) may be expressed as:

E

∞ m =0



( p x , p y ) = E0



∫ ∫ exp( −[t

2 x

+ t y2 ]) exp( −ip xt x − ip y t y )dt x dt y

−∞ −∞

or, if we separate the variables: ∞



−∞

−∞

Em∞=0 ( px , p y ) = E0 ∫ exp(−t x2 ) exp(−ip xt x )dt x ∫ exp(−t y2 ) exp(−ip y t y )dt y ∞

we make use of a table integral183,

∫ exp(−t

2

)dt = π and obtain:

−∞ ∞

∫ exp(−t

−∞



2 x

) exp(−ip xt x )dt x = exp(− p / 4) ∫ exp(−t x2 − ip xt x − i 2 p y2 / 4)dt x = 2 x

−∞



exp(− p x2 / 4) ∫ exp(−[t x + ip x / 2]2 )dt x = π 1 / 2 exp(− p x2 / 4) −∞



and, similarly, for y-component:

∫ exp(−t

2 y

) exp(−ip y t y )dt y = π 1 / 2 exp(− p y2 / 4)

−∞

The resulting Fourier transform of Eq. (A.5) now simplifies to a Gaussian distribution: E m∞=0 ( p x , p y ) = E 0π exp(− p 2 / 4)

(A.6)

where p 2 = p x2 + p y2 . We may notice that ∞

E ( px , p y ) = E0 ∫ ∞ m



∫ ([i ∂ ∂p

−∞ −∞

x

− ∂ ∂p y ])|m| exp(−[t x2 + t y2 ]) exp(−ip x t x − ip y t y )dt x dt y

A.FOURIER TRANSFORM OF LAGUERRE-GAUSSIAN BEAMS 125 and express the higher order Laguerre-Gaussian modes (m>0) from m=0 mode by differentiating with respect to px and p y : Em∞ ( p x , p y ) = ([i ∂ ∂p x − ∂ ∂p y ])|m| Em∞=0 ( p x , p y )

(A.7)

For example, for m=1 we obtain from Eq.’s (A.6) and (A.7): E m∞=1 ( p x , p y ) = E 0 (π / 2)(−ip x + p y ) exp(− p 2 / 4)

(A.8)

2. Fourier transform of a truncated Laguerre-Gaussian beams For a half-beam, truncated by the plane (x < 0), the field in the near-field is zero for x < 0 and is given by Eq. (A.2) for x > 0. The Fourier-transform may be expressed as: ∞

EH∞ , m ( px , p y ) = E0 ∫ 0



∫ ([t

x

+ it y ])| m| exp(−[t x2 + t y2 ]) exp(−ip xt x − ip y t y )dt x dt y (A.9)

−∞

For a Gaussian beam (m=0) we again separate the variables: ∞



0

−∞

EH∞ , m = 0 ( p x , p y ) = E0 ∫ exp(−t x2 ) exp(−ip xt x )dt x ∫ exp(−t y2 ) exp(−ip y t y )dt y

and for the x-coordinate (dimension-less coordinate tx) obtain: ∞

∫ exp(−t 0



1/ 2



2 x

) exp(−ip xt x )dt x = ∫ exp(−t x2 )[cos( p x t x ) − i sin( p xt x )]dt x = 0

/ 2) exp(− p / 4) − i ( p x / 2) exp(− p x2 / 4)1 F1 (1 / 2;3 / 2; p x2 / 4) 2 x

where 1 F1 (α ; γ ; z ) is the degenerate hypergeometric function183. Integration over the ycoordinate was conducted earlier in the derivation of Eq. (A.6). The resulting expression for the k-space representation of a half-Gaussian field is given by: EH∞ ,m=0 ( p x , p y ) = E0 (π / 2) exp(−[ p x2 + p y2 ] / 4)(1 − i ( p x / 2)1 F1 (1 / 2;3 / 2; p x2 / 4))

(A.10)

A.FOURIER TRANSFORM OF LAGUERRE-GAUSSIAN BEAMS 126 For a truncated Laguerre-Gaussian beam of arbitrary m we may again use the method of derivatives, expressed by Eq. (A.7). For m=1 we obtain from Eq. (A.10): E H∞ ,m =1 ( p x , p y ) = E 0 (π / 2) exp(− p 2 / 4) [(−ip x / 2)(1 − ip x / 2)1 F1 (1 / 2;3 / 2; p x2 / 4) + (k y / 2)(1 − ip x / 2)1 F1 (1 / 2;3 / 2; p x2 / 4) + (1 / 2)1 F1 (1 / 2;3 / 2; p x2 / 4) + i (1 − ip x / 2)1 ∂F1 (1 / 2;3 / 2; p x2 / 4) ∂p x ]

Using relations (see 183, 9.213) d 1F1 (α , γ , z ) / dz = (α / γ ) F1 (α + 1, γ + 1, z ) ,

df (t 2 ) / dt = 2t (df ( p ) / dp) p = t 2 , and an algebraic simplifications, we find: EH∞ , m =1 ( p x , p y ) = E0 (π / 4) exp(−[ p x2 + p y2 ] / 4) [[1 − i ( p x + ip y )(1 − ip x / 2)]1 F1 (1 / 2;3 / 2; p x2 / 4)

(A.11)

+ p x (i + px / 2) / 31 F1 (3 / 2;5 / 2; p x2 / 4)]

For m=0, 1 hypergeometric functions may be conveniently replaced by Erf functions183. In that notation Eq. (A.10) simplifies to E H∞ ,m=0 ( p x , p y ) = E0 (π / 2) Erfc (ip x / 2) exp(− p 2 / 4)

(A.12)

where Erfc(z) is a complementary error function. For m=1 Eq. (A.11) becomes: E H∞ ,m=0 ( p x , p y ) = E0 (π / 2) Erfc (ip x / 2) exp(− p 2 / 4)

(A.13)

BIBLIOGRAPHY

127

BIBLIOGRAPHY 1.

Ginzburg, V.L. and L.P. Pitaevskii, "On the theory of Superfluidity". Sov. Phys. JETP, 1958. 7.(858): p. 4.

2.

Green, S., Fluid Vortices. Fluid Mechanics and Its Applications, ed. S. Green. 1995, Dordrecht: Kluwer Academic Publishers.

3.

Lugt, H.J., Vortex Flow in Nature and Technology. 1983, New York: Wiley & Sons.

4.

Vasnetsov, M. and K. Staliunas, Optical Vortices. Horizons in World Physics, ed. M. Vasnetsov and K. Staliunas. Vol. 228. 1999, Huntington, NY: Nova Science.

5.

Siegman, A.E., Lasers. 1986, Mill Valley, CA: University Science.

6.

Nye, J.F. and M.V. Berry, "Dislocations in wave trains". Proc. R. Soc. London Ser. A, 1974. 336: p. 165-190.

7.

Baranova, N.B., et al., "Speckle-inhomogeneous field wavefront dislocations". Pis'ma Zh. Eks. Teor. Fiz., 1981. 33: p. 206-210.

8.

Nye, J.F., "Optical caustics in the near field from liquid drops". Proc. R. Soc. London Ser. A, 1978. 361: p. 21-41.

BIBLIOGRAPHY 9.

128

Deykoon, A.M., et al., "Nonlinear optical catastrophe from a smooth initial beam". Opt. Lett., 1999. 24: p. 1224-1226.

10.

Swartzlander, G.A., Jr., and C.T. Law, "Optical vortex solitons observed in Kerr nonlinear meida". Phys. Rev. Lett., 1992. 69: p. 2503-2506.

11.

Snyder, A.W., L. Poladian, and D.J. Michell, "Parallel spatial solitons". Opt. Lett., 1992. 17: p. 789-791.

12.

Swartzlander, G.A., et al., "Optical transistor effect using an optical vortex soliton". Laser Physics, 1995. 5: p. 704-709.

13.

Gahagan, K.T., G.A. Swartzlander, and Jr., "Optical vortex trapping of particles". Opt. Lett., 1996. 21: p. 827-829.

14.

Simpson, N.B., L. Allen, and M.J. Padgett, "Optical tweezers and optical spanners with Laguere-Gaussian modes". J. Mod. Opt., 1996. 43: p. 2485-2491.

15.

Swartzlander, G.A. and Jr., "Peering into darkness with a vortex spatial filter". Opt. Lett., 2001. 26: p. 497-499.

16.

Palacios, D.M., et al., "Observed scattering into a Dark Optical Vortex Core". Phys. Rev. Lett., 2002. 88: p. 103902.

BIBLIOGRAPHY 17.

129

Marathay, A., Elements of Optical Coherence Theory. 1982: John Wiley & Sons, Inc.

18.

Mandel, L. and E. Wolf, Optical Coherence and Quantum optics. 1995, New York: Cambridge U. Press.

19.

Maleev, I.D., G.A. Swartzlander, and Jr., "Composite Optical Vortices". J. Opt. Soc. Am. B., 2003. 20(6): p. 1169.

20.

Kopf, U., Visualization of phase-objects by spatial filtering of laser speckle photographs. Optik, 1972. 36(5): p. 592-5.

21.

Courant, R. and D. Hilbert, "Methods of mathematical physics". 1953-1962, New York: Interscience Publishers.

22.

Rozas, D., Z.S. Sacks, and J. G. A. Swartzlander, "Experimental observation of fluid-like motion of optical vortices". Phys. Rev. Lett., 1997. 79: p. 3399-3402.

23.

Allen, L., et al., "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes". Phys. Rev. A, 1992. 45: p. 8185-8189.

24.

Bryngdahl, O., "Radial- and Circular-Fringe Interferograms". J. Opt. Soc. Am., 1973. 63: p. 1098.

BIBLIOGRAPHY 25.

130

Bryngdahl, O. and W.H. Lee, "Shearing Interferometry in Polar Coordinates". J. Opt. Soc. Am., 1974. 64: p. 1606.

26.

Coullet, P., L. Gill, and F. Rocca, "Optical vortices". Opt. Comm., 1989. 73: p. 403-408.

27.

Berry, M.V., "Singularities in Waves and Rays", in Physics of Defects, Les Houches Sessions XXXV, R. Balian, M. Kleman, and J.-P. Poirier, Editors. 1981, North Holland: Amsterdam.

28.

Wright, F.J., "Wavefront Dislocations and Their Analysis Using Catastrophe Theory", in Structural Stability in Physics, W. Guttinger and H. Eikemeier, Editors. 1979, Springer-Verlag: Berlin.

29.

Vaughan, J.M. and D.V. Willets, "Interference Properties of a Light Beam having a Helical Wave Surface". Opt. Comm., 1979. 30: p. 263.

30.

Vaughan, J.M. and D.V. Willets, "Temporal and interference fringe analysis of TEM01 laser modes". J. Opt. Soc. Am., 1983. 73: p. 1018-1021.

31.

White, A.G., et al., "Interferometric measurements of phase singularities in the output of a visible laser". J. Mod. Opt., 1991. 38: p. 2531-2541.

32.

Abramochkin, E. and V. Volostnikov, "Beam transformations and nontransformed beams". Opt. Comm., 1991. 83: p. 123-135.

BIBLIOGRAPHY 33.

131

Pismen, L.M. and A.A. Nepomnyashchy, "On interaction of spiral waves". Phys. D, 1992. 54: p. 183-193.

34.

Lugiato, L.A., C. Oldano, and L.M. Narducci, "Cooperative frequency locking and stationary spatial structures in lasers". J. Opt. Soc. Am. B., 1988. 5: p. 879888.

35.

Lippi, G.L., et al., "Interplay of linear and nonlinear effects in the formation of optical vortices in a nonlinear resonator". Phys. Rev. A, 1993. 48: p. R40434046.

36.

Grantham, J.W., et al., "Kaleidoscopic Spatial Instability: Bifurcation of Optical Transverse Solitary Waves". Phys. Rev. Lett., 1991. 66: p. 1422-1425.

37.

Brambilla, M., et al., "Transverse laser patterns. I. Phase Singularity crystals". Phys. Rev. A, 1991. 43: p. 5090-5113.

38.

Indebetouw, G. and D.R. Korwan, "Model of vortices nucleation in a photorefractive phase-conjugate resonator". J. Mod. Opt., 1994. 41: p. 941-950.

39.

Slekys, G., K. Staliunas, and C.O. Weiss, "Motion of phase singularities in a class-B laser". Opt. Comm., 1995. 119: p. 433-446.

40.

Staliunas, K. and C.O. Weiss, "Nonstationary vortex lattices in large-aperture class-B lasers". J. Opt. Soc. Am. B., 1995. 12: p. 1142-1149.

BIBLIOGRAPHY 41.

132

Heckenberg, N.R., et al., "Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator". Phys. Rev. A, 1996. 54: p. 110.

42.

Firth, W.J. and A. Lord, "Two-dimensional solitons in a Kerr cavity". J. Mod. Opt., 1996. 43: p. 1071-1077.

43.

Weiss, C.O., C. Tamm, and P. Coullet, "Temporal and spatial laser instabilities". J. Mod. Opt., 1990. 37(1825-1837).

44.

Vaupel, M. and C.O. Weiss, "Circling optical vortices". Phys. Rev. A, 1995. 51: p. 4078-4085.

45.

Baranova, N.B., et al., "Wave-front dislocations: topological limitations for adaptive systems with phase conjugation". J. Opt. Soc. Am., 1983. 73: p. 525528.

46.

Kruglov, V.I. and R.A. Vlasov, "Spiral self-trapping propagation of optical beams in media with cubic nonlinearity". Phys. Rev. A, 1985. 111: p. 401.

47.

Abramochkin, E. and V. Volostnikov, "Spiral-Type Beams". Opt. Comm., 1993. 102: p. 336.

BIBLIOGRAPHY 48.

133

Liu, S.R. and G. Indebetouw, "Periodic and chaotic spatiotemporal states in a phase-conjugate resonator using a photorefractive BaTiO 3 phase-conjugate mirror". J. Opt. Soc. Am. B., 1992. 9: p. 1507.

49.

Smith, C.P., et al., "Low energy switching of laser doughnut modes and pattern recognition". Opt. Comm., 1993. 102: p. 505-514.

50.

Darsht, M.Y., et al., "Formation of an isolated wavefront dislocation". Sov. Phys. JETP, 1995. 80: p. 817.

51.

Beijersbergen, M.W., et al., "Helical-wavefront laser beams produces with a spiral phaseplate". Opt. Comm., 1994. 112: p. 321-327.

52.

Bazhenov, V.Y., M.V. Vasnetsov, and M.S. Soskin, "Laser beams with screw dislocations in their wavefronts". JETP Lett., 1990. 52: p. 429-431.

53.

Heckenberg, N.R., et al., "Generation of optical phase singularities by computergenerated holograms". Opt. Lett., 1992. 17: p. 221-223.

54.

Roux, F.S., "Diffractive optical implementation of rotation transform performed by using phase singularities". Appl. Opt., 1993. 32: p. 3715-3719.

55.

Roux, F.S., "Branch-point diffractive optics". J. Opt. Soc. Am. A, 1994. 11: p. 2236-2243.

BIBLIOGRAPHY 56.

134

Sacks, Z.S., et al., "Holographic formation of optical-vortex filaments". J. Opt. Soc. Am. B., 1998. 15: p. 2226-2234.

57.

Tamm, C. "The field pattern of a hybrid mode laser: detecting the "hidden bistability of the optical phase pattern". in Proceedings of the Workshop on Quantitative Characterisation of Dynamical Complexity in Nonlinear Systems. 1989.

58.

Tamm, C. and C.O. Weiss, "Bistability and optical switching of spatial patterns in laser". J. Opt. Soc. Am. B., 1990. 7: p. 1034-1038.

59.

Beijersbergen, M.W., et al., "Astigmatic laser mode converters and transfer of orbital angular momentum". Opt. Comm., 1993. 96(1-3): p. 123-132.

60.

Arecchi, F.T., et al., "Vortices and defect statistics in two-dimensional optical chaos". Phys. Rev. Lett., 1991. 67(27): p. 3749-52.

61.

Ackermann, T., E. Kriege, and W. Lange, Opt. Comm., 1995. 115: p. 339.

62.

Ilyenkov, A.V., et al., "Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO/sub". Appl. Phys. B, 1996. 62: p. 465.

BIBLIOGRAPHY 63.

135

Ilyenkov, A.V., et al., "Birth, evolution and annihilation of phase singularities in the propagation of a laser beam passed through a self-focusing strontium barium niobate crystal". J. Nonlin. Opt. Phys. Mater., 1997. 6: p. 169.

64.

Soskin, M.S. and M.V. Vasnetsov, "Nonlinear singular optics". Pure Appl. Opt., 1998. 7: p. 301-311.

65.

Law, C.T., G.A. Swartzlander, and Jr., "Optical vortex solitons and the stability of dark soliton stripes". Opt. Lett., 1993. 18: p. 586-588.

66.

Law, C.T., G.A. Swartzlander, and Jr., "Polarized optical vortex solitons: instabilities and dynamics in Kerr nonlinear media". Chaos, Solitons & Fractals, 1994. 4: p. 1759-1766.

67.

Luther-Davies, B., R. Powles, and V. Tikhonenko, "Nonlinear rotation of threedimensional dark spatial solitons in a Gaussian laser beam". Opt. Lett., 1994. 19: p. 1816.

68.

Swartzlander, G.A. and Jr., "Optical Vortex Solitons", in Spatial Solitons, S. Trillo and W. Torruellas, Editors. 2001, Springer.

69.

McDonald, G.S., K.S. Syed, and W.J. Firth, "Optical vortices in beam propagation through a self-defocusing medium". Opt. Comm., 1992. 94: p. 469476.

BIBLIOGRAPHY 70.

136

Jeng, C.-C., et al., "Partially incoherent optical vortices in self-focusing nonlinear media". Phys. Rev. Lett., 2004. 92: p. 043904.

71.

Mcdonald, G.S., K.S. Syed, and W.J. Firth, "Dark spatial soliton break-up in the transverse plane". Opt. Comm., 1993. 95: p. 281-288.

72.

Staliunas, K., "Vortices and Dark Solitons in the Two-dimensional nonlinear Schrodinger Equation". Chaos, Solitons & Fractals, 1994. 4: p. 1783-1796.

73.

Chiao, R.Y., et al., "Solitons in Quantum Nonlinear Optics", in Frontiers in nonlinear optics: the Serge Akhmanov memorial volume, H. Walther, Editor. 1992, A. Hilger: Bristol, UK. p. 151-182.

74.

Wright, E.M., R.Y. Chiao, and J.C. Garrison, "Optical Anyons: Atoms Trapped on Electromagnetic Vortices". Chaos, Solitons & Fractals, 1994. 4: p. 1797-1803.

75.

Tikhonenko, V., et al., "Observation of vortex solitons created by the instability of dark soliton stripes". Opt. Lett., 1996. 21: p. 1129-1131.

76.

Rozas, D., et al., "Propagation dynamics of optical vortices". J. Opt. Soc. Am. B., 1997. 14: p. 3054-3065.

77.

Chen, Y. and J. Atai, "Dynamics of optical-vortex solitons in perturbed nonlinear medium". J. Opt. Soc. Am. B., 1994. 11: p. 2000-2003.

BIBLIOGRAPHY 78.

137

Christou, J., et al., "Vortex soliton motion and steering". Opt. Lett., 1996. 21: p. 1649-1651.

79.

Kivshar, Y.S., et al., "Dynamics of optical vortex solitons". Opt. Comm., 1998. 152: p. 198-206.

80.

Kivshar, Y.S. and X. Yang, "Perturbation-induced dynamics of dark solitons". Phys. Rev. E, 1994. 49: p. 1657-1670.

81.

Kivshar, Y.S. and B. Luther-Davies, "Dark optical solitons: physics and applications". Physics Reports, 1998. 298: p. 81-197.

82.

Scriabin, D.V. and W.J. Firth, "Dynamics of self-trapped beams with phase dislocations in saturable Kerr and quadratic nonlinear media". Phys. Rev. E, 1998. 58: p. 3916-3930.

83.

Kim, G.H., et al., "An array of phase singularities in a self-defocusing medium". Opt. Comm., 1998. 147: p. 131-137.

84.

Neshev, D., et al., "Motion control of ensembles of ordered optical vortices generated on finite extent background". Opt. Comm., 1998. 151: p. 413-421.

85.

Naschie, M.S.E., "Special Issue: Nonlinear Optical Structures, Patterns, chaos". Chaos, Solitons & Fractals, 1994. 4(8/9).

BIBLIOGRAPHY 86.

138

Parts, U., et al., "Vortex Sheet in Rotating Superfluid 3He-A". Phys. Rev. Lett., 1994. 72(24): p. 3839-3845.

87.

Scott, R.G., et al. "Creation of solitons and vortices by Bragg reflection of a BoseEinstein condensate in an optical lattice". in Quantum Electronics and Laser Science Conference. 2002. Washington, DC: OSA Technical Digest.

88.

Baizakov, B.B., B.A. Malomed, and M. Salerno, "Multidimensional solitons in periodic potentials". Europhys. Lett., 2003. 63(5): p. 642-8.

89.

Kevrekidis, P.G., et al., "Vortices in a Bose-Einstein condensate confined by an optical lattice". J. Phys. B, 2003. 36(16): p. 3467-76.

90.

Bazhenov, V.Y., M.S. Soskin, and M.V. Vasnetsov, "Screw dislocations in light wavefronts". J. Mod. Opt., 1992. 39: p. 985-990.

91.

Khonina, S.N., et al., "The phase rotor filter". J. Mod. Opt., 1992. 39: p. 11471154.

92.

Heckenberg, N.R., et al., "Laser beams with phase singularities". Opt. Quant. Electr., 1992. 24: p. S951-S962.

93.

Basistiy, I.V., et al., "Optics of light beams with screw dislocaitons". Opt. Comm., 1993. 103: p. 422-428.

BIBLIOGRAPHY 94.

139

Indebetouw, G., "Optical vortices and their propagation". J. Mod. Opt., 1993. 40: p. 73-87.

95.

Colstoun, F.B.d., et al., "Transverse modes, vortices and vertical-cavity surfaceemitting lasers". Chaos, Solitons & Fractals, 1994. 4: p. 1575-1596.

96.

Nye, J.F., "Natural focusing and Fine Structure of Light". 1999, Bristol and Philadelphia: Institute of Physics Publishing.

97.

Smith, C.P., et al., "Experimental Realisation and Detection of Optical Vortices", in Optical Vortices, M. Vasnetsov and K. Staliunas, Editors. 1999: Huntington, NY.

98.

Basistiy, I.V., M.S. Soskin, and M.V. Vasnetsov, "Optical Wavefront Dislocations and Their Properties". Opt. Comm., 1995. 119: p. 604.

99.

Harris, M., C.A. Hill, and J.M. Vaughan, "Optical helices and spiral interference fringes". Opt. Comm., 1994. 106: p. 161-166.

100.

Heckenberg, N.R., et al., Optical Fourier transform recognition of phase singularities in optical fields", in From Galileo's "Occhialino" to Optoelectronics", P. Mazzoldi, Editor. 1992, World Scientific. p. 848-852.

101.

Young, T., "A Course of Lectures on Natural Philosophy and the Mechanical Arts". 1807, London: Royal Institution.

BIBLIOGRAPHY 102.

140

Fresnel, A.J., Oeuvres complètes d'Augustin Fresnel. Vol. 1. 1866, Paris: Imprimerie Impériale.

103.

Fresnel, A.J., Oeuvres complètes d'Augustin Fresnel. Vol. 2. 1868, Paris: Imprimerie Impériale.

104.

Fresnel, A.J., Oeuvres complètes d'Augustin Fresnel. Vol. 3. 1870, Paris: Imprimerie Impériale.

105.

Cittert, P.H.v., "Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder einer Linse beleuchteten Ebene". Physica, 1934. 1: p. 201210.

106.

Cittert, P.H.v., "Kohaerenz-Probleme". Physica, 1939. 6: p. 1129-1138.

107.

Zernike, F., The Concept of Degree of Coherence and Its Application to Optical Problems. Physica, 1938. 5: p. 785-795.

108.

Hopkins, H.H., "The Concept of Partial Coherence in Optics". Proc. Roy. Soc., 1951. A208: p. 263-277.

109.

Hopkins, H.H., "On the Diffraction Theory of Optical Images". Proc. Roy. Soc., 1953. A217: p. 263-277.

BIBLIOGRAPHY 110.

141

Parrent, G.B. and Jr., "On the propagation of Mutual Coherence". J. Opt. Soc. Am., 1959. 49: p. 787-793.

111.

Wolf, E., "A Macroscopic Theory of Interference and Diffraction of Light from Finite Sources, I: Fields with a Narrow Spectral Range". Proc. Roy. Soc., 1954. A225: p. 96-111.

112.

Wolf, E., "A Macroscopic Theory of Interference and Diffraction of Light from Finite Sources, II: Fields with a Spectral Range of Arbitrary Width". Proc. Roy. Soc., 1955. A230: p. 246-265.

113.

Roman, P. and E. Wolf, "Correlation Theory of Stationary Electromagnetic Field, Part I: The Basic Field Equations". Nuovo Cimento, 1960. 17: p. 477-490.

114.

Roman, P. and E. Wolf, "Correlation Theory of Stationary Electromagnetic Field, Part II: Conservation Laws". Nuovo Cimento, 1960. 17: p. 477-490.

115.

Brown, R. Hanbury, and R.Q. Twiss, "Correlation between Photons in Two Coherent Beams of Light". Nature, 1956. 177: p. 27-29.

116.

Mandel, L., "Fluctuations of Photon Beams and Their Correlations". Proc. Roy. Soc., 1958. 72: p. 1037-1048.

117.

Dicke, R.H., "Coherence in Spontaneous Radiation Processes". Phys. Rev., 1954. 93: p. 99-110.

BIBLIOGRAPHY 118.

142

Glauber, R.J., "The Quantum Theory of Optical Coherence". Phys. Rev., 1963. 130: p. 2529-2539.

119.

Glauber, R.J., "Coherent and incoherent States of the Radiaiton Field". Phys. Rev., 1963. 131: p. 2766-2788.

120.

Sudarshan, E.C.G., "Equivalence of Semiclassical and Quantum Mechanical Description of Statistical Light Beams". Phys. Rev. Lett., 1963. 10: p. 277-279.

121.

Klyshko, D.N., Photons and Nonlinear Optics (Translation of: Fotony i nelineinaia optika.). 1988, New York: Gordon and Breach.

122.

Mandel, L. and E. Wolf, "Coherence Properties of Optical Fields". Rev. Mod. Phys., 1965. 37: p. 231–287.

123.

Perina, J., Coherence of light. 1971, London: Windsor House.

124.

Goodman, J.W., Statistical Optics. 1985, New York: John Wiley and Sons.

125.

Gbur, G., T.D. Visser, and E. Wolf, "Anomalous Behavior of Spectra near Phase Singularities of Focused Waves". Phys. Rev. Lett., 2002. 88(1): p. 013901.

126.

Popescu, G. and A. Dogariu, "Spectral Anomalities at Wave-Front Dislocations". Phys. Rev. Lett., 2002. 88: p. 183902.

BIBLIOGRAPHY 127.

143

Carter, W.H. and E. Wolf, "Coherence properties of lambertian and nonlambertian sources". J. Opt. Soc. Am., 1975. 65(9): p. 1067-1071.

128.

Carter, W.H. and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources". J. Opt. Soc. Am., 1977. 67(6): p. 785-796.

129.

Wolf, E. and W.H. Carter, "Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources". J. Opt. Soc. Am., 1978. 68(7): p. 953-964.

130.

Carter, W.H. and E. Wolf, "Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory". Opt. Acta, 1981. 28(2): p. 227-244.

131.

Carter, W.H. and E. Wolf, "Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. II. Radiation from isotropic model sources". Opt. Acta, 1981. 28(2): p. 245-259.

132.

Schell, A.C., "The Multiple Plate Antenna". 1961, Doctoral dissertation (Massachusetts Institute of Technology) (unpublished), Sec. 7.5.

133.

Schell, A.C., "Technique for the Determination of the Radiation Pattern of a Partially Coherent Aperture". IEEE Trans. Antennas and Propag., 1967. AP-15: p. 187.

BIBLIOGRAPHY 134.

144

Steinle, B. and H.P. Baltes, "Radiant intensity and spatial coherence for finite planar sources". J. Opt. Soc. Am., 1977. 67(2): p. 241-247.

135.

Baltes, H.P., B. Steinle, and G. Antes. "Radiometric and correlation properties of bounded planar sources". in Coherence and Quantum Optics IV. 1978. Rochester, NY: Plenum press.

136.

Carter, W.H. and M. Bertolotti, "An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals". J. Opt. Soc. Am., 1978. 68: p. 329-333.

137.

Wolf, E. and E. Collett, "Partially coherence sources which produce the same far-field intensity distribution as a laser". Opt. Comm., 1978. 25: p. 293-296.

138.

Foley, J.T. and M.S. Zubairy, "The directionality of Gaussian Schell-model beams". Opt. Comm., 1978. 26: p. 297-300.

139.

Ansari, N.A. and M.S. Zubairy, "Second-harmonic generation by a Gaussian Schell-model source". Opt. Comm., 1986. 59: p. 385-390.

140.

Seshadri, S.R., "Partially coherent Gaussian Schell-model electromagnetic beams". J. Opt. Soc. Am. A, 1999. 16(6): p. 1373-1380.

141.

Li, Y. and E. Wolf, "Radiation from anisotropic Gaussian Schell-model sources". Opt. Lett., 1982. 7(6): p. 256-258.

BIBLIOGRAPHY 142.

145

Collett, E. and E. Wolf, "Is complete coherence necessary of the generation of highly directional light beams?" Opt. Lett., 1978. 2: p. 27-29.

143.

Gori, F., G. Guattari, and C. Padovani, "Modal expansion for J0-correlated Schell-model sources". Opt. Comm., 1987. 64(4): p. 311-316.

144.

Seshadri, S.R., "Average characteristics of a partially coherent Bessel-Gauss optical beam". J. Opt. Soc. Am. A, 1999. 16(12): p. 2917-2927.

145.

Gori, F., et al., "Partially coherent sources with helicoidal modes". J. of Mod. Opt., 1998. 45: p. 539.

146.

Bouchal, Z. and J. Perina, "Non-diffracting beams with controlled spatial coherence". J. of Mod. Opt., 2002. 49: p. 1673.

147.

Ponomarenko, S.A., "A class of partially coherent beams carrying optical vortices". J. Opt. Soc. Am. A, 2001. 18: p. 150.

148.

Bogatyryova, G.V., et al., "Partially coherent vortex beams with a separable phase". Opt. Lett., 2003. 28: p. 878.

149.

Schouten, H., et al., "Phase singularities of the coherence functions in Young's interference pattern". Opt. Lett., 2003. 28: p. 968.

BIBLIOGRAPHY 150.

146

Gbur, G. and T. Visser, "Coherence vortices in partially coherent beams". Opt. Comm., 2003. 222: p. 117.

151.

Maleev, I.D., et al., "Spatial correlation vortices: Theory". submitted to J. Opt. Soc. Am. B, 2004.

152.

Maleev, I.D., G.A. Swartzlander, and Jr., "Propagation of Spatial Correlation Vortices". Opt. Comm., 2004. to be submitted.

153.

Maleev, I.D., et al., "Spatial Correlation Vortices: Detection". submitted to Opt. Lett., 2004.

154.

Palacios, D.M., et al., "Spatial Correlation Singularity of a Vortex Field". Phys. Rev. Lett., 2004. 92(14): p. 143905.

155.

Fresnel, A.J., "Diffraction of light". A. Mem. de l'Acad. Sci, 1826. 5: p. 339.

156.

Whitman, A.M. and M.J. Beran, "Beam spread of laser light propagating in a random medium". J. Opt. Soc. Am., 1970. 60(12): p. 1595-602.

157.

Siegman, A.E., "New developments in laser resonators". Proc. SPIE Int. Soc. Opt. Eng., 1990. 1224: p. 2-14.

158.

Simon, R., N. Mukunda, and E.C.G. Sudarshan, "Partially coherent beams and a generalized ABCD-law". Opt. Comm., 1988. 65(5): p. 322-328.

BIBLIOGRAPHY 159.

147

Visser, T.D., A.T. Friberg, and E. Wolf, "Phase-space inequality for partially coherent optical beams". Opt. Comm., 2001. 187: p. 1-6.

160.

Rayleigh and Lord, Scientific Papers. Phil. Mag., 1897. 43: p. 259.

161.

Berry, M.V., "Coloured phase singularities". N. J. of Phys., 2002. 66: p. 1-14.

162.

Berry, M.V., "Exploring the colours of dark light". N. J. of Phys., 2002. 74: p. 114.

163.

Palacios, D., G.A. Swartzlander, and Jr. "White Light Vortices". in The OSA Annual Meeting & Exhibit". 2002. Washington D. C.: Optical Society of America.

164.

Palacios, D.M., G.A. Swartzlander, and Jr., "An Achromatic Null in the Presence of Fractional Vortices". to be submitted to J. Opt. Soc. Am. B, 2004.

165.

Swartzlander, G.A., Jr., and J. Schmit, "Temporal Correlation Vortices and Topological Dispersion". submitted to Phys. Rev. Lett., 2004.

166.

Gahagan, K.T., G.A. Swartzlander, and Jr., "Simultaneous Trapping of Low-index and High-Index Microparticles With an Optical Vortex Trap". J. Opt. Soc. Am. B., 1999. 16: p. 533-537.

BIBLIOGRAPHY 167.

148

He, H., et al., "Direct Observation of Transfer of Angular Momenum to Absorptive Particles from a Laser Beam with a Phase Singularity". Phys. Rev. Lett., 1995. 75: p. 826-829.

168.

He, H., N.R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical particle trapping with higher-order doughnut beams produced using high-efficiency computer generated holograms". J. Mod. Opt., 1995. 42: p. 217-223.

169.

Friese, M.E.J., et al., "Optical angular-momentum transfer to trapped absorbing particles". Phys. Rev. A, 1996. 54: p. 1593-1596.

170.

Simpson, N.B., et al., "Mechanical equivalent of spin and orbital angular momentum of light: an optical spanner". Opt. Lett., 1997. 22: p. 52-54.

171.

Saleh, B.E.A. and M.C. Teich, Fundamentals of Photonics. Wiley Series in Pure and Applied Optics, ed. J.W. Goodman. 1991, New York: Wiley & Sons.

172.

Dholakia, K., et al., "Second-harmonic generation and the orbital angular momentum of light". Phys. Rev. A, 1996. 54: p. R3742-3745.

173.

Berzanskis, A., et al., "Conversion of topological charge of optical vortices in a parametric frequency converter". Opt. Comm., 1997. 140: p. 273-276.

174.

Soskin, M.S., et al., "Topological Charge and Angular Momentum of Light Beams Carrying optical Vortices". Phys. Rev. A., 1997. 56: p. 4064-4075.

BIBLIOGRAPHY 175.

149

Berzanskis, A., et al., "Sum-frequency mixing of optical vortices in nonlinear crystals". Opt. Comm., 1998. 150(1-6)(372-80).

176.

Berzanskis, A., et al. "Parametric amplification of an optical vortex". in Proceedings-of-the-SPIE-The-International-Society-for-Optical-Engineering. 1997. SPIE-Int. Soc. Opt. Eng: Wroclaw, Poland.

177.

Di-Trapani, P., et al., "Observation of optical vortices and J/sub 0/ Bessel-like beams in quantum-noise parametric amplification". Phys. Rev. Lett., 1998. 81: p. 5133-6.

178.

Maleev, I.D., et al. "Violation of conservation of topological charge in optical down-conversion". in Quantum Electronics and Laser Science Conference. 1999. Washington, D. C.: OSA Technical Digest.

179.

Mair, A., et al., "Entanglement of the orbital angular momentum states of photons". Nature, 2001. 412: p. 313.

180.

Basistiy, I.V., et al., "Experimental observation of rotation and diffraction of "singular" light beam". Proc. SPIE Int. Soc. Opt. Eng., 1998. 3487: p. 34.

181.

Padgett, M.J. and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes". Opt. Comm., 1995. 121: p. 36-40.

BIBLIOGRAPHY 182.

150

Nourrit, V., J.-L.d.B.d.l. Tocnaye, and P. Chanclou, "Propagation and diffraction of truncated Gaussian beams". JOSA A, 2001. 18(3): p. 546-556.

183.

Gradshteyn, I.S. and I.M. Ryzhik, "Table of Integrals, Series, and Products". 1965: Academic Press, Inc.

184.

Rozas, D., G.A. Swartzlander, and Jr., "Observed rotational enhancement of nonlinear optical vortices". Opt. Lett., 2000. 25: p. 126-128.

185.

Press, W., et al., "Numerical recipes in C++". 2002: Cambridge press.

186.

To alleviate the problem of diverging in our numerical calculations when the intensity vanishes, we truncated to zero when the intensity fell below a 2% threshold.

187.

Swartzlander, G.A. and Jr., "Optical Vortex Filaments", in Optical Vortices, M. Vasnetsov and K. Staliunas, Editors. 1999, Nova Science: Huntington, NY.

188.

Maleev, I.D., et al., "Spatial Correlation Vortices: Detection". to be submitted to Opt. Lett., 2004.

189.

Mertz, L., "Transformations in Optics". 1965, New York: Wiley.

190.

Thompson, B.J. and E. Wolf, J. Opt. Soc. Am., 1957. 47: p. 895.

BIBLIOGRAPHY 191.

151

Allen, L., S.M. Barnett, and M.J. Padgett, "Optical Angular Momentum". 2003, Philadelphia: Institute of Physics Publishing.

192.

Deutsch, I.H., "The Quantum Theory of Paraxial Optical Fields, and Two-Photon Bound-States in a Self-Focusing Kerr Medium", in Physics. 1992, University of California at Berkeley: Berkeley.

193.

Arlt, J., et al., "Parametric down-conversion for light beams possessing orbital angular momentum". Phys. Rev. A, 1999. 59: p. 3950–3952.