Chapter 1

Introduction to linear singular optics

I.I. Mokhun

Chernivtsy University, Ukraine

1.1. IntRoduction

Propagation of a coherent radiation through inhomogeneous media with random fluctuations of local optical characteristics results in forming the optical wave that is characterized by random temporal and spatial distributions of its parameters, such as intensity, phase and, in general case, state of polarization. The fields formed in such a way are referred to as speckle fields [1,2].

While the parameters of speckle-fields are described by the complex functions of general form, one can expect that diverse peculiarities (or singularities) are inherent in them, both point-like and extended ones. Such singularities and stationary points are interconnected in the form of peculiar nets [3-26]. These nets, as skeletons, constitute the field structure, and information on the characteristics of these sets provides the feasibilities for predicting (at least qualitatively) the behavior of the field at any its point.

It must be emphasized that the fundamentals of this approach have been developed in the epoch-making book “Natural focusing and fine structure of light” by J. Nye issued in 1999 [3]. The problems elaborated in details in Ref. 3 are only mentioned in passing in this chapter, which is intended as some (successful, as the authors hopes) addendum and sequel of this outstanding study.

On this reason, the main attention is paid here to the decisive role of singularities of various kinds in formation of the structure of electromagnetic field. The author hopes that his sight of the problems of singular optics, as well as of the advances of the singular optical approach in solving the problems of optics and of the role of the nets of singularities as the skeletons of electromagnetic field, will be of interest for many readers involved in modern optics.

For the sake of observability, the author restricts the consideration by rigorously monochromatic waves. The optical singularities arising in optical waveguides, nonlinear media etc. are not considered also, while this is the subject for another comprehensive book.

Note in conclusion that the number of publications devoted to singular optics grows as avalanche in last years, and on this reason the list of references may seem as too short, especially in regard to the concluding subsection dealing with the singularities of the Poynting vector. Nevertheless, the list of references contains the key publication concerning the main aspects of singularities in electromagnetic fields.

1.2. Basic notions of scalar singular optics

1.2.1. Phase vortices

It is known [12,28] that uniformly polarized wave of general form with frequency , , being propagating in a free space, obeys the wave equation:

, (1.1)

where , and – wave length of radiation. Among various solutions of this equation exists the solutions, for which the following relation is satisfied:

, (1.2)

where is a complex amplitude of the field, and is the unitary vector determining its state of polarization. For that, satisfies the Laplace equation:

. (1.3)

The solutions of Eq. (1.3) include all analytical functions with complex variables, the simplest of which is of the form:

_. (1.4)

For that, satisfies the wave equation (1.1). Eq. (1.4) can be easily transformed to the form that is generally accepted as the definition of complex amplitude:

_, (1.5)

where is the amplitude, and is the spatial phase. It follows from Eq. (1.5) that and can be regarded as the polar coordinates with the origin at . It is seen that the amplitude of this wave approaches zero, while the phase is undetermined as . In other words, one observes a phase singularity at the point . Such wave formations were called as the wave front dislocation or optical vortex [3-17,27,29-39]. Taking into account that below we will consider vortices of various nature, let us refer to these vortices as the phase vortices.

a b с

Figure 1.1. Phase map of an isotropic vortex. Levels of gray correspond to changes of a phase: (a) and (b) correspond to the signs + and – In Eq. (1.4), respectively; (c) shows changes of a phase under circumference of the vortex (solid and dashed lines for the positive and negative vortices, respectively).

a b с

Figure 1.2. Phase map of an anisotropic vortex.

Phase distribution (a phase map) in the vortex vicinity defined by Eq. (1.4) is represented in Figure 1.1. One can see that under circumference of the vortex centre the phase changes linearly, being growing or decreasing depending on the sign + or – in Eq. (1.4). In paper [12,27] such vortices were called as isotropic ones. In general case (as one can see from Figure 1.1) the equiphase line associated with zero magnitude of a phase does not coincide with -axis of the laboratory coordinates with the origin at the vortex centre. Magnitude of a phase shift, , hereinafter is referred to as the initial phase of an isotropic vortex. Note, that the phase change shown in Figure 1.1(c) can be obtained from the phase distribution illustrated in Figure 1.1(a) by a simple rotation of the coordinates at the angle .

a b

Figure1.3. Replacing of an anisotropic vortex by the isotropic one.

Of course, Eq. (1.4) is seldom applicable for generic optical field. The area where the phase changes obey linear approximation is referred to as the vortex core [3,4]. As a rule, behavior of a phase within the vortex core is governed by any nonlinear rule (see Figures 1.2 and 1.3) that corresponds to appearance the real coefficients at and in Eq. (1.4), see Figure 1.2. The most general description of the vortex structure is achieved through additional rotation of the phase structure of a vortex, as it is shown in Figure 1.3(a). One can see from this figure that the line of zero phase does not coincide with the -axis, and the angle between equiphase lines and differs from 90^о. Such vortices are called as anisotropic ones [12,27].

There are general features of the phase changes under circumference of the vortex:

1. Phase surface in the vortex vicinity is the clockwise or the counterclockwise helicoid. This is the only case when the phase change by under circumference of the vortex takes place.

2. Phase change in the function of a polar angle is monotonic (Figure 1.3(b)). That is why, any isotropic vortex can be put in correspondence to the given anisotropic one [25,26]. Let us call this isotropic vortex as the characterizing vortex. Phase change of the characterizing vortex is depicted in figure by the dished line. Note that the maximal phase difference of the genuine vortex, , and of the characterizing vortex does not exceed . Thus, following the Rayleigh criterium, we can conclude that any anisotropic vortex can be replaced, within accuracy , by an isotropic vortex with specified magnitude of the initial phase .

3. By passing the vortex centre, the phase undergoes a jump by . As a consequence, change of the phase at adjacent areas of the magnitude is the same, .