Singularities are places at which some quantities become undefined. For example, as shown in Fig. 1.3-1, the center of a color wheel is a color singularity at which the color becomes undefined. The basic reasons study singularities is because of their ubiquity and structural stability [Berr80]. For optical waves, there are mainly two kinds of singularities being concerned: phase singularities and polarization singularities [Nye99]. The survey of these singularities has becomes a very modern area of interest in contemporary physics and is named singular optics [SV01].
Generally, phase singularities [Berr98] are points in plane and lines in space at which intensity vanishes and the phase of complex scalar wave field become undefined. In this work only 2D complex scalar field is concerned, which stand either for 2-D quantum wave function or for transverse modulus of light beam. It is convenient to introduce the mathematical form
( , )x y R x y( , ) i I x y ( , )
. (1.3.1)
with R x y and ( , )( , ) I x y to be real. By defining ( , )x y R x y2( , )I2( , )x y and ( , )x y arg[ ( , )] x y , the scalar field can rewrite as
( , )x y ( , ) exp[ ( , )]x y i x y . (1.3.2)
The positions at which R x y and ( , )( , ) I x y simultaneously equal to zero such that the amplitude ( , )x y vanishes and the phase ( , )x y becomes undefined are the phase singularities. These nodal points in 2D plane are analogous to crystal dislocation and are also referred as phase dislocation [NB74].
Another important quantity related to the phase is the probability current density
which is generally given by
Substituting Eq. (1.3.2) in to Eq. (1.3.3), the probability current density can be alternatively expressed as the gradient of the phase ( , )x y
( , ) ( , ) ( , )
According to fundamental calculus, the curl of j
will be zero at all positions except for the phase singularities. Hence, phase singularity is also termed as vortex for the circulating current density around it. The vortices have been involved in a wide variety of coherent phenomena such as superconducting films [MFDM03], superfluid [MFDM03], Bose-Einstein condensate [MAH+99], microwave billiards [ŠHK+97], quantum ballistic transport [BSS02], and liquid crystal films [dGP93].
One important characteristic of a phase singularity is its topology charge (also named as winding number or dislocation strength) defined by
1 1
, where C is arbitrary closed loop containing only one singularity inside. The charge is positive (negative) if the phase circulates counterclockwise (clockwise). A crucial topological property of singularities is the sign rule which indicates that the charge of the neighboring singularities on a constant phase contour must have opposite signs [Freu95].
Due to the underlying analogy between optical momentum density and the probability current density (See Appendix B for a more detailed discussion), the phase singularity of the amplitude distribution of light beam manifest itself as optical vortex
[VS99]. Consequently, optical vortices is intimately related to optical OAM [SGV+97]. As revealed in last section, optical OAM has attracted much interest because of the wide applications, such as atom trapping [KTS+97], optical tweezers [MRS+99], optical spanner [SADP97], and quantum information [MVWZ01].
In singular optics to generate optical vortex is one of the predominant topics.
Typically, optical vortices can be generated by passing a fundamental-mode Gaussian beam through such as cylindrical-lens mode converters [BAv+93], holograms [HMS+92], spiral phase plates [BCKW94], axicons [KKS+07], uniaxial crystals [VSF+06], and glass wedges [YAC+07]. Besides, spontaneous formations of optical vortices in laser system have also been reported in solid-state lasers [CL01, OC09], Na2 laser [BBL+91] and proton-implanted vertical-cavity surface-emitting laser (VCSEL) [SO99]. The mechanism of the vortex formation in proton-implanted VCSEL is due to transverse mode locking, assisted by the laser nonlinearity, of nearly degenerate Lagurre-Gaussian modes [SO99]. Different from proton-implanted VCSEL, the near-field transverse modes of oxide-confined VCSELs were shown to be analogous to closed-quantum-billiard wave functions [HCLL02, CHLL03a, CLS+07, CSCH08], which are purely real and contain only zero phases. However, transverse field becomes complex as soon as it propagates out of the VCSEL cavity [CYC+09]
and contains intricate vortex structure, as will be shown in chapter4.
In addition to phase singularity of complex scalar waves, the singularities at which the orientations of a real vector field become undefined are the so-called vector field singularities [Denn01], or vector singularities [Freu01] in brief. In terms of mathematical expression, a 2-D real vector field can be written as
( , ) x( , ) ( , ) ˆx y ˆ V x y V x y a V x y a
y. (1.3.6) The vector singularities are the positions at which and equal to zero simultaneously such that the orientation angle determined by the angle function
( , )
V x yx V x yy( , )
( , )x y angle V x y V x y[ x( , ), y( , )]
(1.3.7)
becomes undefined. The topological charge of a vector singularity given by
1 1
is called Poincaré index of zero [Denn01], where the contour C should be a very small path around the singularity. The vector singularities with Poincaré index to be 1 can be categorized into vortices, sources and sinks, and saddles. Fig. 1.3-2 (a)-(c) show the distributions of the real vector field around a vortex, saddle, and source, respectively. The contour plots of orientation angles functions of the vector fields shown in Fig. 1.3-2 (a)-(c) are depicted in Fig. 1.3-2 (a’)-(c’). It can be seen that both vortex and source have their Poincaré indices to be 1 and the Poincaré index of a saddle is . In fact the probability current density is one kind of the most familiar vector fields. The locations in phase function
1
( , )x y
correspond to the vector singularities of current density j x y( , )
are called critical points. The critical points of phase giving rise to vortices, sources and sinks, and saddles in current density are singularities, extrema, and saddle. Assume the vector fields shown in Fig. 1.3-2 (a)-(c) are probability current densities of some wave functions. We depict the corresponding phase structures of the wave functions, which containing phase singularity, maximum, and saddle, in Fig. 1.3-3 (a)-(c), respectively. In conclusion, these critical points of scalar function become crucial as some vector field is expressed as the gradient of the scalar function.
Vector singularities have also been involved in a wide variety of physics. For optical waves, vector singularities are isolated, stationary points in a plane at which the orientation of the electric vector of a linearly polarized real vector field becomes undefined [Freu01]. The features of the vector singularities have been experimentally observed in laser modes with the interrelated behavior of spatial
structures and polarization states [Gil93, VKMR01, LCH07, Erdo92, PTMA97]. In this work, the vector singularities embedded in the near filed patterns of VCSELs will be analyzed in an unambiguous way [CHLL03b, CSL+07].
Fig. 1.3-1. A color wheel. At the center the color becomes undefined.
Fig. 1.3-2. (a)-(b) Vector fields with vortex, saddle, and source, respectively;
(a’)-(b’) The corresponding orientation angle function of vector fields shown in (a)-(b).
(a) (a’) 2π
(b)
(c)
(b’)
(c’)
0 2π
0 2π
0
Fig. 1.3-3. (a)-(c) The phase structures of singularities, saddle, and extremum.
The gradient of these phases will result in the vector fields shown in Fig. 1.3-2 (a)-(c), respectively.
(a)
(b)
0 2π
0 2π
(c) 2π
0