1.2.4. Generation of vortices using computer-generated holograms

Obviously, a device for vortex generation can be performed as a hologram, whose transmittance is like to the pattern shown in Figure 1.5 [37-39].

Let us consider interference of the vortex beam, , with a plane reference wave, , Let us assume the vortex beam to be isotropic vortex with the positive topological charge:

, (1.9)

and the plane wave to be incident at angle in respect to the -axis:

. (1.10)

Then, intensity of the resulting field is of the form:

. (1.11)

Further, let us assume that the fixed at the photoplate intensity distribution possesses an amplitude transmittance:

, (1.12)

where the coefficients are associated with nonlinearity of a holographic recording. Taking into account Eq. (1.11), one can rewrite Eq. (1.12) in the form:

, (1.13)

where the coefficients also reflects nonlinearity of a holographic recording, and is the number of the diffraction order of a hologram.

It follows from Eq. (1.13) that when a hologram is illuminated with a plane wave propagating in the directions , the vortex beams with the topological charges are reconstructed. To say, at the second diffraction order ( ) the vortex with the doubled negative topological charge is reconstructed [37,38].

1.3. Vortices and phase structure of a scalar field

1.3.1. Sign principle

We have analyzed in p.1.2.4 the events of birth and annihilation of topological units of the field phase. It was stated that such events are obey by pair principle. This principle, being quite fundamental, does not answer the question, what vortex among the pair of two born vortices bears positive (or negative) topological charge. The statement filling this gap, known as the sign principle, has been formulated in Ref. [13].

Note that the coordinates of vortices (amplitude zeroes) can be found as the solutions of the following system [3,8,9]:

. (1.14)

These equations determine two sets of equiphase lines, and , respectively. Cross-sections of these lines (hereinafter referred to as lines) are just the loci of the vortices of the field.

Figure 1.6. In correspondence with the sign of and , the numbers denote sub-areas of the field, within which a phase changes do not exceed .

The essence of the principle is straightforward, as it is seen from Figure 1.6. This figure shows the area of a random field divided into sub-areas, within which the phase is constant (within ). Sub-areas are bounded by the lines, which are the solutions of Eqs. (1.14). Referring to Figure 1.6, one can formulate the sign principle in the following way: Adjacent vortices lying at some line and positioned at the crossings of these lines must be of opposite topological charge.

Note, the notion of a line of constant phase with its specific value is relative to a certain extent in optics, while due to rapid change of a phase in time the only phase difference among mutually coherent waves has relevant physical sense; only this quantity can be registered in practice using, to say, interferometric techniques.

It is obvious, the vortices of opposite signs may be connected by a bundle of equiphase lines, whereas the vortices of the same sign are connected by the only line passing the saddle point (see figure).

One can reformulate the sign principle: If the adjacent vortices can be connected by an equiphase line, which does not pass the saddle, then such vortices are of opposite sign of topological charge; in other cases, such vortices are of the same sign. This quite general principle is applicable for analysis of arbitrary wave field.

As a simple application of this principle, we now consider a part of the or line, where two adjacent vortices, ( ) and ( ), with signs (+) and (–), respectively, occur. If a new pair of vortices ( ), ( ) arises between vortices and, then the sign principle requires that the new configuration must be of the form ( )( )( )( ). Note, the pair principle presumes also another configuration, ( )( )( )( ), which is forbidden by the sign principle.

Note that the pair principle is the rule justified for any isolated point of the wave function; as so, the pair principle is the local one. At the same time, the sign principle establishes the connection between different points of the wave function and, as such, is the global principle. One can see that the sign principle can not be reduced to the pair principle. On the other hand, numerous consequences of the pair principle can be obtained as the direct consequences of the sign principle.

Let us formulate three main consequences of the sign principle:

1. The sign of any specified vortex of the wave field automatically determines (prescribes) the signs of all other vortices of this field.

2. If the sign of any vortex is changed, then the signs of all other vortices are changed also.

3. The sign of the first vortex, which was born due to the evolution of a wave field, pre-determined the signs of the vortices, which would be born in future.