1.3.2. Phase speckles. “breathing” of phase speckles

L

Figure 1.7

et us introduce the notion of a “phase speckle”. [25]. We use this term (see Figure 1.7) to designate a spatial structural element of the field of phase bounded by the , lines and by crossings of them, which form one-folded closed line. It is important, the location of the , lines is determined within to the constant phase factor. The nodes of the speckle and the stationary points of a phase are the fixed elements of the phase speckle. As the phase factor common for the field as a whole, one can use the temporal phase factor, , or the spatial phase

Figure 1.8. «Breathing» of phase speckles. Computed-simulated phase speckles. Subsequent fragments differ by 0.05 . Lines and differing in levels of gray are shown. Corresponding to the interval where the phase changes (within the each speckle, the phase changes within p/2), the phase speckles depicted by levels of gray and by white. Growing the mean phase within the speckle corresponds to density of gray.

factor, , if the wave propagates along z-axis. Depending of this choice, the , lines oscillate in space between two saddles. Figure 1.8 illustrates such “breathing” of the phase speckles. Phase speckles corresponding to the field at far zone were obtained by computer simulation.

Phase changes from Figure 1.8(a) to Figure 1.8(f) with the step 0.05 . The highest speed of the shift of the , lines in a function of changing the constant phase of a field is observed within the region of the stationary points of the field of a phase, while the lowest speed is observed within the regions of considerably large gradients of a phase, i.e. between the saddles.

1.3.3. Birth of vortices

The results of experimental investigation of the processes of vortex birth and annihilation are represented in Ref. 9. Namely density of wave-front dislocations was measured as a function of the distance from the scattering object.

F or clearness, let us use a scattering object introducing a pure phase modulation (kind of ground glass). Immediately behind the scattering object, modulation of the field is a phase-only; so, the vortices are absent in the boundary field of the object [40]. Obviously, the vortices are born due to multi-beam interference at near field, where the field becomes modulated both in phase and in amplitude.

T

Figure 1.9

he process of generation of vortices and growing their number occurs jump-like, see Figure 1.9, so that at the beginning of the Fresnel zone density of zeroes (within to angular divergence of the field) achieves its resulting magnitude, which is not changed up to infinity [9].

F

Figure 1.10

igure 1.10 shows schematically the process of development of a three-dimensional speckle pattern. So-called shadow region, , where a speckle pattern is formed is the zone of vortex generation. For that, one can not conclude that evolution of vortices within the Fresnel zone is the same as a far-field propagation of vortices. The body of a speckle within Fresnel zone is of the more or less “true“ ellipsoidal form. Speckle sizes (including longitudinal ones) are finite and increase as the observation plane moves toward the Fraunhoffer zone. For that, the boundary of a speckle, viz. its cross-section can be described as single-fold curves of more or less complex form. The lines (trajectories) of amplitude zero lie along a speckle boundary, being the closed single-fold curves also. As the plane of analysis moves along -axis, one observes the birth, propagation and annihilation of vortices [41-43]. The vortex is born at the point where the line of amplitude zero winding a speckle body is tangent to this plane (the point in Figure 1.10). Further moving the plane of analysis along -axis results in appearance of two vortices, and , up to the point where the line of zero amplitude is tangent to this plane again (point ). Here the vortices annihilate. In other words, evolution of vortices within the Fresnel zone (with constant mean density of vortices) can be considered as the process of altering events of birth and annihilation of vortex pairs. In this sense, the dynamics of vortex nets within the Fresnel zone radically differs from the evolution of vortices at far field, where the lines of amplitude zeroes are closed at infinity. That is why, starting from some instant (from some position of the plane of analysis at -axis), the number of events of annihilation and birth of new pairs of vortices decreases, and, in the end, these events cease. Stationary (within the angle factor) nets of vortices are observed at larger magnitudes of .

Thus, the vortices resulting from multi-beam interference do not annihilate at far field.

In spite of the mechanism of vortex appearance seems to be obvious (it is reduced to interference of partial wavelets), peculiarities of this process is not quite clear. From our point of view, the physical nature of this phenomenon might be represented in the following form. As soon as the of vortices are born, on the whole, between the boundary object field and Fresnel zone where the phase modulation predominates, the field forming under propagation of the wave within this region can be determined using the wave front approximation (see Appendix 1.1). So, the only adjacent areas of the wave front are involved into creation of the vortices, whereas the contributions from the wavelets form any removed sources are negligible. In this case, the event of appearance of a vortex can be interpreted as interference of limited (small) number of partial wavelets with approximately equal intensities and with enough smooth wave fronts.

The principles of forming the dislocation nets under interference of three plane waves and Gaussian beams are considered in Ref. 44-46. In practice, however, representation of interfering wavelet as plane or Gaussian ones must be performed with known caution. Further we consider appearance of vortices as the result of interference of the waves of general form.