POLARIZATION SINGULARITIES

Polarization Singularities In Paraxial Vector Fields: Morphology And Statistics



Polarization Singularities In Paraxial Vector Fields: Morphology And Statistics

Polarization singularities in paraxial vector fields: morphology and statistics zero, the phase of the field is undetermined or singular. Singular optics is concerned with the description and classification of the different kinds of singularities that can occur in wave fields [2, 3]. Examples of such singularities are the zeros of intensity that are found in focused fields. In real-valued, two-dimensional vector fields, the orientation of the vector is singular wherever the vector vanishes.

Such singularities of the Poynting vector field in two-dimensional geometries are studied in Refs. -. Complex-valued vector fields can display singularities of the vector components. Examples of these are singularities of the longitudinal component of the electric field in strongly focused, linearly polarized beams. Recently, the two-point correlation functions that describe spatially partially coherent light were shown to posses singularities as well - . All types of singularities mentioned above can be created or annihilated when a system parameter, such as the wavelength of the field, is smoothly varied.

At every point in a time-harmonic electromagnetic field, the end point of the electric field vector traces out an ellipse as time progresses. The polarization is said to be singular at points where this ellipse degenerates into a circle (at so-called C-points) or into a line (at so-called L-lines). Polarization singularities in wave fields are described in Refs. , and -.

Because of their use in, for example, optical trapping, the properties of focused, radially polarized beams have been studied extensively in the past few years. The electric field in the focal region of such a beam has two non-zero parts, namely a radial component and a longitudinal component. The creation and annihilation of phase singularities of these field components has been described in Ref. It the present paper the rich polarization behavior of focused, radially polarized fields is analyzed. It is shown that the focal region contains different kinds of polarization singularities such as L -lines, stars, monstars, lemons, and V-points. Their interrelation is examined, and it is demonstrated how polarization singularities can be created or annihilated when, e.g., the semi-aperture angle of the focusing system is changed.

Consider an aplanatic focusing system L, as depicted in Fig. 1. The system has a focal length f and a semi-aperture angle ?. The origin O of a right-handed cartesian coordinate system is taken to be at the geometrical focus. A monochromatic, radially polarized beam is incident on the system. The electric and magnetic fields at time t at position r are given by the expressions

E(r, t) = Re[e(r)exp(-i?t)] , (1)

H(r, t) = Re[h(r)exp(-i?t)] , (2)

respectively, where Re denotes the real part. The longitudinal component ez and the radial component e??of the electric field at a point P = (?P, zP) in the focal region are given by the equations

where w0 is the spot size of the beam in the waist plane, which is assumed to coincide with the entrance plane of the focusing ...