1.3.4. Appearance of wave front dislocations as a result of interference of waves with simple phase surfaces

Let us consider interference of the minimal number, viz., two, arbitrary coherent waves, and [4,47,48]. Phases of these waves, and , and their amplitudes, and , are such that the each of two waves can be represented within the wave fronts approximation (see Appendix 1.1) at any observation plane perpendicular to -axis determining the main direction of propagation of the waves. In other words, we assume propagation of the waves and within the region of interest to be diffractionless. At point of the observation plane the amplitudes are approximately equal. In this case, one can look for the solution of equation in the form , viz., the isophotes ( the lines of equal fields amplitudes).

The condition of appearance of an isolated amplitude zero of the resulting field at point has the form:

. (1.15)

Let us analyze the resulting field formed by the waves and . We introduce a local coordinates, , with the origin at the -th amplitude zero. Directions of -axis and -axis coincide. Let us decompose the resulting field in Maclaurin’s series on degrees of and , holding the linear terms alone. In this case, the resulting field is represented in the form:

, (1.16)

where , , , , , .

Tangent of the phase of the resulting field in the vicinity of the -th amplitude zero is described as:

. (1.17)

It follows from Eq. (1.17) that the field in the vicinity of amplitude zero is none other than a screw dislocation of a wave front. The sign of the vortex is determined by the ratio of and , i.e. by gradients of the amplitudes, as well as by gradient of phases of the interfering beams.

Let us analyze the connection among the vortex “phases” in the resulting field. By analogy with the initial phase of an isotropic vortex, cf. subsection 1.2.1, we refer to the vortex phase as any constant phase in the vicinity of the vortex center, which can be determined for the each point of this region as a constant “support” for the changing component of a phase. Similarly to the case of an isotropic vortex, this phase just determines the intensity of the interference pattern in the vicinity of amplitude zero, viz., “dark”, “bright” or “gray” forklet. Let us introduce the parameter , which specifies the position of the point at the isophote, viz., the line of constant intensity. The resulting field along this line is described by the relation:

. (1.18)

At the point , where the isophotes intersect the th minimum of the interference pattern, the argument of the exponential factor determines the vortex phase. Taking into account Eqs. (1.16) and (1.18), one finds that this phase is described by the relation:

(1.19)

Note, that the coordinate can be associated with the phase surface of any of two fronts. Obviously, the argument of the exponential factor remains unchanged under circumference of the point at the plane at small distances from the point . Phase difference at the adjacent minima of the interference pattern is determines as:

. (1.20)

It follows from Eq. (1.20) that the phase difference at two adjacent points equals . Imposing the reference wave on such field, one observes the interference forklets corresponding to the mentioned singularities which occur different in brightness, viz., if one of two forklets is formed as bifurcation of a dark interference fringe, than the other one is formed as bifurcation of a bright interference fringe.

Suppose the both waves are plane and are directed in such a manner that the line of intersection of the wave fronts is parallel to the axis, and the wave fronts are inclined in respect to the plane at the equal angles , cf. Figure 1.11. For that, the intensities of the waves are changed in the direction, being the constant along the axis. Then, , , and .

As a result, Eq. (1.17) is transformed to the form:

. (1.21)

It follows from Eq. (1.21) that while , and , the sign of the multiplier to the ratio is determined by the sign of alone, as well as the signs of the vortices. Furthermore, if the intensity of any of two waves is constant, the signs of the vortices are determined by the gradient of an intensity of other wave.

T his statement is confirmed by computer simulation, see Figure 1.12, where the result of interference of two quasi-plane waves with approximately equal intensities and the reference beam is illustrated. The vertical fringes of the pattern correspond to the resulting field with the vortices arisen. The horizontal fringes result from a superposition of this field with the reference beam. The character of the changing intensity of the interfering waves is sketched in Figures 1.12(e)-(g). The isophotes are depicted by the arrows in Figures 1.12(a)-(c).

One can observe in Figure 1.12(a) the vortices of the same sign at the adjacent minima. The corresponding zones are depicted by rectangles with the characters A and B. The phases of the vortices at the adjacent minima differ by . It is seen clearly that the forklet depicted by the rectangle A is “black”, being, to say, inserted in the “white” forklet depicted by the rectangle B. The changing intensity is illustrated in Figure 1.12(e).

Experimentally obtained chain of such vortices is illustrated in Figure 1.13.

I

Figure 1.11

n Figures 1.12(b),(c) one can observe two sets of the interference forklets of opposite signs in correspondence with changing intensity of the waves shown in Figure 1.12(f). The vortices with opposite signs of the topological charge are depicted by the rectangles A and B. Note, the peculiarities of the interference pattern are gradually blurred, viz., the forklets of opposite signs become hardly discriminated, as the distance between the vortices diminishes. At last, the vortices annihilate in Figure 1.12(g).

Figure 1.12

Figure 1.13

For that, a singular point of the phase arises, which can be interpreted as the “edge dislocation of zero length” ( ). It can be shown that the topological charge of the regionincluding such a defect of the field is , and the index is . Thus, annihilation of the vortices, in respect of Eq. (1.7), is realized in two steps:

. (1.22)

Namely, the vortices annihilate firstly, and the edge dislocation of zero length arises. For that, two saddles merge, and the “folding” of a phase is smoothed after this event:

. (1.23)

A

a b c

Figure 1.14

s a result, the annihilation process can be regarded as the chain: destruction of the vortices as more “powerful” singularities beating non-zero topological charge, and then final disappearance of the field singularities [49-52]. Note, such a chain does not contradict to the conservation law for the total topological charge and index, while these quantities are constant at the each stage.

Interference formation of the vortices is modeled experimentally using any interferometer, including a shearing interferometer. Using a shearing interferometer, one can implement the interference converter of a smooth beam into singular one [53].

We used a glass wedge (the angle between sides being of magnitude) as a shearing interferometer. Illuminating such a wedge by the He-Ne laser beam, one observes in reflected radiation an interference pattern with large enough period (up to several millimeters), i.e. the interfering beams are almost collinear. The back side of the wedge is covered by a thin aluminum layer, cf. Figure 1.14(a), with the coefficient of reflection approaching 100%. The layer covering the front side of the wedge was prepared in such a manner, that the coefficient of reflection changed gradually from 0 to 100%. For that, a transmittance changed in the direction perpendicular to the interference fringes. As a result, the beams 1 and 2 reflected from the front and the back sides of the wedge, respectively, acquired modulation in intensity, cf. Figure 1.14(b). In such a manner, the conditions necessary for the interference formation of vortices were provided. Figure 1.14(с) illustrates the results of testing of the transformed beam.