The role of caustics in formation of network of amplitude

zeroes for partially developed speckle field

O.V. Angelsky1, P.P. Maksimyak1, A.P. Maksimyak1, S.G. Hanson2 and Yu.A. Ushenko1

1 Correlation Optics Dept, Chernivtsi University, 2 Kotsyubinsky Str., Chernivtsi, 58012 Ukraine

2 Optics and Fluid Dynamics dept., RISØ National Laboratory, P.O. Box 49, DK-4000 Roskilde,

Denmark.

Abstract. The topology of a partially developed speckle field is studied using interference

techniques. Amplitude and phase structures in the vicinity of caustics for coherent radiation

field scattered at a surface with large inhomogeneities are investigated. It has been confirmed

that the caustics are inalienable components in the scenario of amplitude zeroes networks

formation for a coherent field scattered by a rough surface with large inhomogeneities. It has

been shown that formation of interference forklets in the field gives evidence of changes in

the field topology, being the diagnostic sign of transition from a planar Fraunhofer diffraction

pattern to a three-dimensional pattern of a diffraction catastrophe.

OCIS codes: 240.5770, 290.0290, 120.3180, 120.4630, 170.6960, 180.1790.

1. Introduction

Among the problems that are solved successfully within the concept of singular

optical, one must emphasize:

� the techniques for generation of isolated singularities (also called wave front dislocations,

vortices, or amplitude zeroes) in the field [1, 2];

� the topology of the network of amplitude zeroes for the speckle field at the model

(qualitative) level [3-5];

� the optical techniques for diagnostics of singularities for the field [6, 7];

� the use of singularities for the field in solving application problems, such as manipulating

small particles (i.e. optical traps and tweezers) [8, 9].

The interest in the study of local amplitude and phase structures of the field in the

vicinity of singularities, not only for monochromatic radiation field but also for polychromatic

fields, has considerably increased recently [3, 10-12]. Papers [13, 14] devoted to the study of

the scenario and mechanisms of nucleation and evolution of singularities in the vicinity of

caustics have appeared.

Despite for reasons of fundamental nature, such investigations are stimulated by

problems in applied diagnostic. As a matter of fact, the successful solving of diagnostic

problems by determination of statistical or fractal parameters of the objects via analysis of

large areas of the scattered radiation field (networks of vortices) is hampered without

knowing the mechanisms of nucleation of amplitude zeroes. Actually, the extensive

diagnostic usage of the networks of stable structure, such as amplitude zeroes, is a matter of

interest for the nearest future. Besides, initial successful attempts to use a totality of these

structural formations for the field in diagnostic problems have just been reported [15]. At last,

knowledge of the structural amplitude and phase parameters of the field at micro- and

nanoscale (subwavelength) levels is of undoubted importance for solving the problems of

nanotechnology [16, 17].

Nucleation of amplitude zeroes for the coherent radiation field takes place only if a

phase variance of inhomogeneities of the rough surface (with which the beam interacts)

exceeds unity. In this case, partial signals with phase differences exceeding π take part in

formation of the scattered field, and, as a consequence, amplitude zeroes are nucleated [18]. A

rough surface with such height parameters can be considered as the generating source for

singularities. The areas of sharp focusing of partial signals (caustics) also take part in the

development of singularities for the speckle field. That is why the interest in local amplitude

and phase structures of the field is evident [19].

There are considerations showing that amplitude zeroes of the field and the

accompanying networks are promising in solving diagnostic problems. The network of

amplitude zeroes constitutes a singular skeleton for the field. The reasons for this conclusion

are given in [4]; the primary reasons being:

� a phase structure for a Gaussian speckle field can be reconstructed, in principle, from the

network of amplitude zeroes;

� an alteration of the sign of a single amplitude zero unavoidably results in alteration of the

signs for the entire network;

� the number of amplitude zeroes for the field equals the number of speckles considered as

ensemble averages;

� the amplitude zeroes are located at the periphery of the speckles, where the speed of a

phase change attains its maximum.

Besides, it is convenient to use the lines of zero amplitude for description of the

complex pattern, including the speckle field. They are structurally important because topology

dictates that they are endless lines in space. Moreover, they are structurally stable: i.e. for

small changes of the controlling parameters they simply move, but retain their identities; they

cannot disappear [13].

One can suppose that the scenario of nucleation and evolution of singularities depend,

apart from anything else, on the structure of the singularity generating object, for example, in

the case of random and fractal surfaces [15].

The purpose of this paper is to study and demonstrate the mechanisms of nucleation

and evolution of singularities for a coherent radiation field scattered by a surface with large

inhomogeneities, at the zone, in which partial signals are focused, i.e. in the vicinity of

caustics.

2. Interference study of the scattered field structure

To determine the loci of amplitude zeroes for the field, we use the interference

technique [6], which is now of the widest use for detection of optical wave front singularities.

Amplitude zeroes are revealed by typical interference forklets (or bifurcations of interference

fringes). Besides, we determine the loci of amplitude zeroes by solving the system of

equations:

[ ][ ]

Re ( , ) 0,

Im ( , ) 0.

A x y

A x y

=

= (1)

using the software approved in [15].

In this study we consider nonfractal random surfaces (NRS). A generator of random

normally distributed numbers controlled the height distribution of the NRS. The objects are

formed by 900×900 pixels. Furthermore, the NRS objects undergo 2-D smoothing when

either a Gaussian or an exponential law is followed with various half-widths for the height-

distribution function. In such a way one obtains quasi-smoothed micro-irregularities of

different transverse scale, which are reproduced by a distribution of pixel values. An example

of a surface modeled in this way is shown in Fig. 1, namely – the smoothed NRS based on an

exponential law over three pixels.

The height-distribution function for the surface irregularities is also represented in

Fig.1 as well as the statistical parameters for the surface.

The example considered above refers to a span of surface irregularities heights, hmax =

2 µm. In the simulations, hmax was changed from 0.1 to 10 µm, which corresponded to a phase

variation from 0.5 to 50 radians.

The following procedure was carried out for calculation of the diffracted field from

the rough surface. Let us consider a transmitting object with a rough surface. This case is

straightforwardly implemented. The approach for the case of a reflecting rough surface is

identical. So, for the transmitting object, φ(x,y) = k(n - 1)h(x,y), while for the reflecting one,

φ(x,y) = 2kh(x,y), where h(x,y) is the relief height of a rough surface, n is the bulk index of

refraction, k = 2π/λ is the wave number, and λ is the wavelength. Throughout our

computations and experiments, λ = 0.633 µm (He-Ne laser) and n = 1.46 (fused quarts).

The amplitude and phase of the field resulting from diffraction of a plane wave at the

phase relief of a rough surface can be calculated by the double Rayleigh-Sommerfeld

diffraction integral [20]:

( ) ( )

[ ]{ }dxdyy))h(x,ζ)(nξ,z,y,R(x,ik

ζ)ξ,z,y,(x,R

yx,A

iλ

z=ζξ,U

1exp

2

−−×

×∫ ∫ , (2)

where A(x,y) is the aperture function that corresponds to the amplitude transmittance of a

rough surface (see Fig. 2), R(x, y; ξ, ζ, z) = [z2 + (x - ξ)2 + (y – ζ)2 ]1/2 is the distance between

the surface point and the observation point, z is the distance between the plane of the object to

the observation plane, and x, y; ξ, ζ are the rectangular Cartesian coordinates at the object

plane and the observation plane, respectively, as shown in Fig. 2. Eq. (2) is applicable to the

field calculations at an arbitrary distance z. In this study we replace integration by summation,

dividing both the object and the field in the observation plane into elementary areas.

The field is computed at various distances from the object. Subsequently, a coherent

reference wave is imposed on the computed field to analyze the field based on the

interference pattern. The period of this is chosen to be several times less than the speckle size.

Thus, one can determine the loci of field singularities and estimate the phase distributions of

the field in the vicinity of the amplitude zeroes. As an example, we show interferograms and

intensity distributions of the field obtained at various distances from a rough surface in Fig. 3.

The observation distances are selected from convenience for qualitative examination of the

scenario of evolution of the field’s structural peculiarities. Fig. 3 (b) shows an interferogram

of the field registered at a distance of m5=z µ . It is seen that at this zone amplitude-phase

modulation of the field takes place, which reveals itself in the corresponding configuration

and in the spatial frequency of interference fringes. In Fig. 3 (c, d) we show the intensity

distribution of the field and the interferogram of the field at a zone of sharp focusing of partial

signals (i.e. the caustics zone). Generally, the row of such zones can be distinguished at the

field scattered by a random rough surface. Whereas the area of caustics is determined by the

relation between the phase variance and the correlations length of inhomogeneities of a rough

surface, partial signals corresponding to various scales of the surface inhomogeneities can be

focused at different distances. The distribution of the corresponding zones of focusing along

the −z coordinate is governed by the distribution of the ratio of the inhomogeneities phase

variance to the associated correlation lengths. As a result, one expects relatively smooth

distribution of such zones along the −z coordinate for a random rough surface, in contrast to

the case of fractal rough surface, where such distribution exhibits pronounced maxima [15].

In Fig. 3 (c – 1, 2, 3) we see the caustic zone closest to the object, where the first-

appearing amplitude zeroes are nucleated. One observes the diffraction maxima decorating

the zones of sharp focusing. The form and loci of these maxima correspond to the form of

caustics. Relative half-period shifts of the interference fringes take place at transition from the

zone of sharp focusing to the neighboring diffraction maximum, cf. Fig. 3 (d-1, 2, 3).

Generally, the observed Fraunhofer diffraction pattern can be specified as “planar,” localized

in the observation plane.

As the observation plane is removed further from the caustics zone, a “planar”

diffraction pattern is transformed into a three-dimensional one, i.e. into diffraction

catastrophe, according to the classification introduced in [14]. Strictly speaking, the

mechanism of formation of singularities involving cross-interference of partial beams is

initiated at this stage. This is seen from Fig. 3(e, f). This stage is of special interest in the

study of the mechanisms for evolution of singularities by the analysis of local phase

distributions into vicinity of caustics of different forms.

Fig. 3(g, h) depicts the intensity distribution and the interferogram of the developed

speckle field, i.e. the stable pattern of the corresponding distribution of speckles and

amplitude zeroes. For comparison, an experimentally obtained interferogram of a speckle

field is shown in Fig. 3(i). We use a rough surface obtained photolitographically as the object

with parameters corresponding to the surface parameters used in the computer simulations.

In general, the represented simulation and experiment confirm that the field scattered

from rough surfaces, the phase variance of which exceeds unity, always contains zones of

sharp focusing, or caustics. This means that caustics are the indispensable attribute of the

scenario of evolution of amplitude zeroes for the speckle field, irrespective of the specific

form of the height distribution function for surfaces with large inhomogeneities.

To demonstrate the viability of the introduced approach, we initially studied known

peculiarities of a speckle field topology. Hence, we have confirmed that each speckle of the

field has an associated amplitude zero in its vicinity, so that the number of amplitude zeroes is

equal, in average, to the number of speckles. This is illustrated in Fig. 4(a, b, c, d), where the

amplitude zeroes are detected interferometrically and, besides, are controlled by a software, as

indicated by small circles. The usage of computer determination of loci of amplitude zeroes is

caused by the fact that as the density of amplitude zeroes grows, the resolution of interference

technique becomes insufficient for carrying out reliable analysis. The presented results

confirm that the number of amplitude zeroes is equal, in average, to the number of speckles.

Interference analysis of the field also permits us to examine a triple forklet at a

distance of 35 µm, which would correspond to an amplitude zero with topological charge 2;

see Fig. 5(b). However, a detailed study conducted by altering the interference angle (in both

sagittal and meridional directions) showed that one deals with two isolated, however very

closely spaced, one-charged vortices; see Figs. 5(c) and 5(d). In fact, a small change in the

interference angle, as well as a small change in the observation plane, results in the decay of a

triple forklet into two conventional forklets; see Fig. 5(e). Thus one can conclude that an

observed triple forklet is caused only by an accidental choice of the observation plane. This

conclusion is in agreement with the well-known finding that high-order amplitude zeroes are

spatially unstable and that they decay into isolated one-charged zeroes of the same sign [1].

Further, we have applied the interference technique for diagnostics of phase saddles in

the field. To reveal such specific areas of the field phase structure, we use the quasi-coaxial

superposition of the coherent reference beam. In this case, a complex phase distribution of the

tested area of the field is transformed into a corresponding intensity distribution that is

different for two orthogonal directions: from maximum through minimum to maximum, and

from minimum through maximum to minimum, see Fig. 6 (a, b, c). One can schematically

imagine a saddle as it is shown in Fig.6 (d). To provide reliable control, we also investigated

an interference pattern arising from off-axis interference of two beams. The absence of

interference forklets in the previously found areas of the loci of the phase saddles confirms

the reliability of the suggested technique.

We have also confirmed that nucleation of an amplitude zero is always accompanied

by the appearance of a phase saddle. This is illustrated in Fig. 7, where the stages of scenario

of singularities nucleation at the area next to the caustics are shown. It is seen that the number

of amplitude zeroes (shown by circles) corresponds to phase saddles (shown by rectangles). It

is important that this rule is valid at each stage of the evolution of singularities, being in

agreement with the prediction by Nye [14].

Using the interference technique we studied the evolution of a pair of anisotropic

dislocations (cf. classification by Freund [4]), namely, the amplitude zeroes shown in Fig. 8.

It has been shown that, during evolution, anisotropy corresponding to a half-period shift of

interference fringes decreases up to its disappearance, Fig. 8 (f).

Within the framework of interference investigation, the effect of so-called clustering

of amplitude zeroes becomes clear. This effect manifests itself in the appearance of areas of

the field, where the spatial density of amplitude zeroes considerably exceeds the average

density [15]. The main prerequisite for this effect is that caustics take part in the formation of

the field or, in other words, a “planar” diffraction pattern is transformed into a three-

dimensional pattern of diffraction catastrophe. In the present case of interest, the caustics

zones play the role of the centers of nucleation and evolution of singularities as the

observation plane moves away from the object, see Fig. 9 (a-g). One can see that the structure

of clusters matches the form of the caustics.

3. Structure of the field in the vicinity of caustics

Note that the approach introduced here allows us not only to look for a scenario of

evolution of considerable fragments of a field, picking up some interesting peculiarities of this

process, but also to investigate the mechanisms of evolution of separate groups of

singularities in the vicinity of caustics of various form, similar to the approach revealed in

[13, 14]. Visual analysis of the computed fragments of a field based on the study of both

intensity distribution and interference patterns at caustics zones permits us to pick up small

areas of a field, whose structure corresponds (in terms of the catastrophe theory) to the fold,

cusp, swallowtail, and so on. Using this possibility, we now investigate the main observable

elementary events causing transformation of the field topology, i.e. we look for the field

evolution from the caustic zone to a far field. Our simulation, being as close as possible to a

real experimental situation, provides discrete analysis of the field at various distances z from

the object. In this case, analysis of the field evolution along the z -coordinate is performed via

comparison of the successive set of interference patterns registered at discrete planes. So, we

compare the patterns illustrating the coordinate distributions of the point of crossing of three-

dimensional lines of zero amplitude with the corresponding planes, see Fig. 10.

It is clear that as the frequency of sampling along the z -coordinate increases, the

ability to visualize and identify the events causing changes in the field topology increases too.

As it has been mentioned, the description of complex structured speckle patterns can be

performed using the lines of zero amplitude. The technique of discrete scanning of the field

along the z -coordinate with subsequent determination of loci of the amplitude zeroes permits

observation of the behavior of the lines of zero amplitude, determination of their form, and

explanation of the events taking place in the field.

It is known that the changes of the field topology are realized as a result of birth or

annihilation of singularities. Such effects are simply detected interferometrically, as seen from

the interferograms in Fig. 11. One observes not only the effect of birth of two pairs of

amplitude zeroes (two dipoles), but also the effect of annihilation of amplitude zeroes from

different dipoles (Fig. 11 (f, g)). As a rule, the number of crossings of the lines of zero

amplitudes with the plane z = const is even, as it is evident from the interferograms shown in

Fig. 4. Besides, the effects of both changing the distances between two dipole dislocations

and rotation of a dipole are found when z is changing (Fig. 12).

Let us consider the mechanisms of the appearance of singularities in the vicinity of

caustics as the registration zone moves away from the focusing zone. We again apply the

interference technique to analyze the field structure. First of all, note that the complex

structure of the singularity-generating object, such as a rough surface with large

inhomogeneities, causes a complex distribution of caustics of various forms, as it can be seen

in Fig. 3 (a). Let us demonstrate some distinctive fragments at this distribution, namely, the

zones of sharp focusing with their neighborhood, see Fig. 13. As an example, consider the

caustics originating from cylindrical lens-like phase element of the rough surface structure

slightly bended at the plane parallel to the plane of observation; this caustic resembles an edge

dislocation. We can observe in Fig. 13 the line of sharp focusing and clearly seen

accompanying (linear) diffraction maxima of the first order, whose form is the same as the

form of the main maximum. Imposing a reference beam, one observes a relative shift of the

interference fringes when crossing the minimum. This confirms that the lines of amplitude

zeroes lie in the plane, and we just observe the “planar” Fraunhofer pattern. The absence of

typical interference forklets shows that screw dislocations are absent in this zone. If we

imagine that the plane of observation is rotated by 900 and a reference wave is imposed, then

we expect to observe an interference forklet, as the amplitude zero line now crosses the plane

of observation. As the plane of observation is removed from the zone of sharp focusing, a

“planar” diffraction pattern evolves into a three-dimensional one, and the lines of zero

amplitude pass from the plane to the volume. The topology of the field changes, which is

evident from the nucleating screw dislocations, detected interferometrically as forklets, cf. Fig

14. Generally, the considered mechanism corresponds to the one described theoretically in

[13], where the passing to a three-dimensional diffraction pattern is accompanied by the

appearance of the corresponding “switches” of arch-like form. Thus, the main principle that

governs the evolution of the lines of amplitude zeroes is confirmed. Each change in the

topology of dislocation on 0=z nucleates a new arch on the 0>z side.

An analogous scenario implementing the mechanism of passing from a “planar”

diffraction pattern to a three-dimensional one is illustrated in Fig. 15. The transition to a

three-dimensional pattern is diagnosed by the detection of interference forklets.

To illustrate the above-mentioned facts, consider Fig. 16. As an example, we show in

Fig. 16 (a, b) the coordinate distribution of amplitude and phase of the field at a zone

preceding the zone of focusing of partial signals. In Fig. 16 (b) the scale of grey corresponds

to the magnitude of the phase. Smooth changing of the grey scale corresponds to a smooth

change of phase, while sharp boundaries (depicted by inserted rectangles) correspond to π -

jump of the phase. Superposition of these two patterns (amplitude and phase) is shown in Fig.

16 (c). This provides comparison of the peculiarities of the coordinate distributions of phase

and amplitude zeroes in the observation plane.

In Fig. 17 (a) we show the joint distribution of the field’s amplitude and phase in the

zone of focusing of partial signals. Note that at some areas depicted by rectangles one

observes absolute collapse in space of amplitude zeroes and lines of π - jumps of the phase.

This result is analogous to the effect of precise coincidence of the lines characterizing the

distributions of real and imaginary parts of the complex amplitude of a field, [ ]ERe and

[ ]EIm , respectively, as it takes place in the zone of focusing of radiation passing a spherical

lens (Airy pattern). So, within the depicted areas, we have the lines of zero amplitude in

zones, where the amplitude zeros coincide with the lines of π - jumps of a phase, localized in

the plane. This can be referred to as the transverse dislocation of the field. At the ends of such

transverse dislocations, where the lines of the phase jumps and amplitude zeroes diverge (go

out from the plane), screw dislocations are found. This is seen from Fig. 17 (b), where

amplitude zeroes of the simulated field are depicted. The same amplitude zeroes associated

with screw dislocations are diagnosed interferometrically, see Fig. 17 (c).

In Fig. 18 (a) we show joint distribution of amplitude and phase of the field at zone

removed at the distance µm=z 10 from the object. Analyzing this figure, one can conclude

that the precise coincidence of amplitude zeroes and π - jumps of the phase are observed only

at isolated points. The number of screw dislocations increases, as it is seen from Fig. 18 (b).

Thus, we observe transition from a “planar” diffraction pattern into a three-dimensional

diffraction catastrophe.

It is known that the lines of zero amplitude are the closed lines, whose projections on a

plane are loops [13]. One can obtain the notion of the behavior of these lines by analyzing

Fig. 19 illustrating the trajectories of amplitude zeroes at the plane ZOY behind the caustics

zone. Traces of the secant planes, shown in the figure as numerated vertical lines, facilitates

the counting of the amplitude zeroes in any plane.

Let us analyze Fig. 19 (a). The plane FF’ is the plane of focusing of partial signals, i.e.

the caustics zone. When crossing the line of zero amplitude with this plane, one detects the

birth of two amplitude zeroes, 1’, 1’’, as a result of change of the field topology. In the course

of further study of the crossings of this line with the observation plane (in our case, with the

planes where amplitude zeroes are counted) we can conclude that amplitude zeroes move in

the horizontal direction, i.e. along the y-axis. In reality, the line of zero amplitude is not

localized in the plane of the figure, but is twisted in three-dimensional space; see Fig. 19 (b).

In this case, we would detect not only a linear motion of the lines of zero amplitude, but also

their rotation as well as through-and-back motion at projection into the plane of observation,

as it is observed in Fig. 12.

Thus, in the evolution of the dislocations from their birth to annihilation one can

observe that, as the observation plane is removed from the point of birth of amplitude zeroes

creating the so-called dipole, the following happens:

� amplitude zeroes can deflect, being at the common plane, which is perpendicular to the

observation plane;

� the pair of amplitude zeroes constituting a dipole, pushing apart, performs rotation or

through-and-back motion on a break trajectory, if the motion is projected to the plane of

observation;

� both the amplitude zeroes of opposite signs constituting one dipole and the amplitude

zeroes of opposite signs of different dipoles can annihilate;

� the appearance of interference forklets in the field behind the caustics zone is the

diagnostic sign of passing from a “planar” Fraunhofer diffraction pattern to a three-

dimensional one, i.e. of the change of the field topology.

4. Conclusion

In summary, we have proposed to use the interference approach for investigating the

scenario of development of singularities in the coherent field scattered by a rough surface

with large inhomogeneities, as a function of the distance to the observation plane. This

approach, based on computer simulations, provides results, which are close to the results of

laboratory experiments. We have demonstrated the feasibility for study of the mechanisms

and peculiarities of the development of complex speckle fields and their amplitude and phase

structure at various scale levels and at various registration zones. We have applied

interference techniques to diagnose phase saddles for the optical field. Known mechanisms

and scenario for formation of partially developed speckle fields with inherent singularities

have been confirmed and demonstrated. The mechanisms of transformation of a developed

speckle field structure resulting in changes of topology have been investigated.

We confirmed that caustics are inalienable components in the scenario for formation

of networks of amplitude zeroes for coherent field scattered by a rough surface with large

inhomogeneities. It has been shown that caustics are the centers of formation of clusters of

amplitude zeroes for a partially developed speckle field.

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Figure captions

Figure 1. Simulated random rough surface exponentially smoothed over three pixels (a), and

the corresponding height distribution function (b); qR and

aR are, respectively, root-mean-

square deviation and arithmetic mean deviation of irregularities from a mean surface line, kS

- asymmetry coefficient, uK - kurtosis coefficient

Figure 2. Notations to the computation of the scattered radiation field scattered at an arbitrary

distance 0z=z from a rough surface.

Figure 3. Intensity distributions (a, c, e, g) and the corresponding interferograms (b, d, f, h) of

the scattered coherent radiation at various distances z from a rough surface: m5=z µ1 (a, b),

m8=z µ2 (c, d), µm=z 153 (e, f), µm=z 5004 (g, h). The most interesting areas are depicted

by rectangles and inserted. One can observe evolution of a diffraction pattern from “planar”

(a, b and, to a certain extent, c, d) to three-dimensional one (e, f to g, h). An experimentally

obtained interferogram from a rough surface is shown in fragment (i).

Figure 4. Intensity distributions (a, c) and the corresponding interferograms (b, d) of the

scattered coherent radiation at distances µm=z 5601 (a, b) and µm=z 7802 (c, d) from a

rough surface. Loci of amplitude zeroes are depicted by small circles. One can see that the

number of speckles is equal, in average, to the number of amplitude zeroes, which are at dark

areas of the field. So, for (a, b) 4444 =x,=s , and for (c, d) 3232 =x,=s , x,s being

average numbers of speckles and amplitude zeroes, respectively.

Figure 5. Fragment of intensity distribution (a) and the corresponding interferograms of the

field (b, c, d, e) obtained at distance µm=z 35 from a rough surface. One can observe

breaking of two-charged amplitude zero (b) into two one-charged amplitude zeroes (d). The

areas of interest are depicted by rectangles. Amplitude zeroes are visualized by interference

forklets (b, c, d, e).

Figure 6. Interference patterns obtained via coaxial superposition of the field with a coherent

reference wave at distances m6=z µ1 (a), m9=z µ2 (b) and µm=z 113 (c) from a rough

surface. Phase saddles are depicted by rectangles. Enlarged images of phase saddles are

shown in the right column. Fragment (d) shows schematically a phase surface in the vicinity

of a saddle point A.

Figure 7. Interferograms of the field at the caustics zone obtained via on-axis (a, c) and off-

axis (b, d) interference. Phase saddles and amplitude zeroes are depicted by rectangles and by

circles, respectively. One can see that the number of phase saddles is equal to the number of

amplitude zeroes.

Figure 8. Evolution of a pair of anisotropic dislocations. Interferograms are obtained at

distances µm=z 7001 (a), µm=z 7102 (b), µm=z 7203 (c), µm=z 7504 (d), µm=z 7705 (e)

and µm=z 8006 (f) from a rough surface. Amplitude zeroes are depicted by circles.

Anisotropy at fragment (a) manifests itself by a half-period shift of interference fringes;

passing to fragment (f) anisotropy vanishes.

Figure 9. Intensity distributions of the field at the caustics zone at distances m6=z µ1 (a),

m7=z µ2 (b), m8=z µ3 (c), m9=z µ4 (d), µm=z 105 (e), µm=z 116 (f), and µm=z 127 (g)

from a rough surface. Amplitude zeroes are depicted by circles. One can observe clustering of

amplitude zeroes at caustics zone. The structure of clusters corresponds to the structure of

caustics.

Figure 10. Scheme of discrete analysis of the field providing determination of behavior of the

lines of zero amplitude. A, B, C, D – discrete plane of analysis of the field propagating along

−z axis. Filled circles depict amplitude zeroes. 1, 1’ and 2, 2’ are pairs of amplitude zeroes,

which are crossings of the lines of zero amplitude with the planes A, B, C, D.

Figure 11. Fragments of interferograms of the field at distances m7=z µ1 (a), m8=z µ2 (b),

m9=z µ3 (c), m=z µ104 (d), µm=z 115 (e), µm=z 126 (f), µm=z 137 (g), and µm=z 148

(h) from a rough surface. Amplitude zeroes are depicted by circles. One can observe birth (c,

d) and annihilation (g) of dislocations (shown by arrows). The events of interest are shown by

arrows.

Figure 12. Fragments of interferograms of the field at distances µm=z 6001 (a), µm=z 6502

(b), µm=z 7003 (c), µm=z 7604 (d), µm=z 7705 (e), µm=z 8006 (f), and µm=z 8107 (g)

from a rough surface. Amplitude zeroes obtained by software are depicted by circles. One can

observe birth of amplitude zeroes, fragment (a), and motion of them along complex trajectory,

including rotation, as the plane of observation is removed from the point of birth of

singularities.

Figure 13. Fragments of intensity distribution (a, b) and the corresponding interferograms (c,

d) at the caustics zone at distance m5=z µ from an object. One can see diffraction maxima as

the lines accompanying the caustics lines, as well as relative half-period shift of interference

fringes along these lines.

Figure 14. Fragment of intensity distribution with the computed amplitude zeroes (shown by

circles) at distance m6=z µ from an object (a), and the corresponding interferogram (b).

Figure 15. Fragments of intensity distribution with the computed amplitude zeroes (shown by

circles) at distance m6=z µ from an object (a, b), and the corresponding interferograms (c,

d).

Figure 16. Amplitude (a) and phase (b) distributions of the field scattered by a rough surface

with large inhomogeneities at distance m5=z µ from the object. Fragment (c) shows the

resulting amplitude and phase distribution of the field resulting from superimposing of the

distributions shown in fragments (a) and (b).

Figure 17. Joint distribution of the field’s amplitude and phase (a), the same with imposed

computed amplitude zeroes (depicted by circles) (b), the associated interferogram of the field

(c) at distance m6=z µ from the object. The areas of interest are depicted by rectangles.

Figure 18. Joint distribution of the field’s amplitude and phase (a) with the computed

amplitude zeroes (b) at distance µm=z 10 from the object.

Figure 19. To explanation of the effect of motion of the amplitude zeroes constituting a

dipole. The plane FF’ place the role of the focal plane of partial focusing. The planes 1, 2, 3,

4 are the secant planes (the planes of observation). Circles depict the points of crossing of the

lines of zero amplitudes with the secant planes, which are the amplitude zeroes (1’, 1’’; 2’,

2’’) at the planes of observation. Coherent wave propagates along z-axis. Fragments (a) and

(b) illustrate two-dimensional and three-dimensional cases, respectively. Fragment (b)

illustrates rotating and through-and-back motion of the dipole.

h, nm

(a)

(b)

Figure 1. Simulated random rough surface exponentially smoothed over three pixels (a), and the corresponding height distribution function (b);

qR and aR are, respectively, root-mean-square

deviation and arithmetic mean deviation of irregularities from a mean surface line, kS -

asymmetry coefficient, uK - kurtosis coefficient

z=z0

Figure 2. Notations to the computation of the scattered radiation field scattered at an arbitrary distance

0z=z from a rough surface.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

12

3

12

3

1

2

3

1

2

3

1 1

2 2

1

2

1

2

(i)

Figure 3. Intensity distributions (a, c, e, g) and the corresponding interferograms (b, d, f, h) of the scattered coherent radiation at various distances z from a rough surface: m5=z m1

(a, b),

m8=z m2 (c, d), ìm=z 153

(e, f), ìm=z 5004 (g, h). The most interesting areas are depicted by

rectangles and inserted. One can observe evolution of a diffraction pattern from “planar” (a, b and, to a certain extent, c, d) to three-dimensional one (e, f to g, h). An experimentally obtained interferogram from a rough surface is shown in fragment (i).

(a) (b)

(c) (d)

Figure 4. Intensity distributions (a, c) and the corresponding interferograms (b, d) of the scattered coherent radiation at distances ìm=z 5601

(a, b) and ìm=z 7802 (c, d) from a rough

surface. Loci of amplitude zeroes are depicted by small circles. One can see that the number of speckles is equal, in average, to the number of amplitude zeroes, which are at dark areas of the field. So, for (a, b) 4444 =x,=s , and for (c, d) 3232 =x,=s , x,s being average numbers of speckles and amplitude zeroes, respectively.

(a) (b) (c) (d) (e)

Figure 5. Fragment of intensity distribution (a) and the corresponding interferograms of the field (b, c, d, e) obtained at distance ìm=z 35 from a rough surface. One can observe breaking of

two-charged amplitude zero (b) into two one-charged amplitude zeroes (d). The areas of interest are depicted by rectangles. Amplitude zeroes are visualized by interference forklets (b, c, d, e).

(a)

(b)

(c)

(d)

A

Figure 6. Interference patterns obtained via coaxial superposition of the field with a coherent reference wave at distances m6=z m1

(a), m9=z m2 (b) and ìm=z 113

(c) from a rough

surface. Phase saddles are depicted by rectangles. Enlarged images of phase saddles are shown in the right column. Fragment (d) shows schematically a phase surface in the vicinity of a saddle point A.

(a) (b)

(c) (d)

Figure 7. Interferograms of the field at the caustics zone obtained via on-axis (a, c) and off-axis (b, d) interference. Phase saddles and amplitude zeroes are depicted by rectangles and by circles, respectively. One can see that the number of phase saddles is equal to the number of amplitude zeroes.

(a) (b) (c)

(d) (e) (f)

Figure 8. Evolution of a pair of anisotropic dislocations. Interferograms are obtained at distances ìm=z 7001

(a), ìm=z 7102 (b), ìm=z 7203

(c), ìm=z 7504 (d), ìm=z 7705

(e) and

ìm=z 8006 (f) from a rough surface. Amplitude zeroes are depicted by circles. Anisotropy at

fragment (a) manifests itself by a half-period shift of interference fringes; passing to fragment (f) anisotropy vanishes.

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 9. Intensity distributions of the field at the caustics zone at distances m6=z m1 (a),

m7=z m2 (b), m8=z m3

(c), m9=z m4 (d), ìm=z 105

(e), ìm=z 116 (f), and ìm=z 127

(g)

from a rough surface. Amplitude zeroes are depicted by circles. One can observe clustering of amplitude zeroes at caustics zone. The structure of clusters corresponds to the structure of caustics.

1

1’

2

2’

1

1’

2

2’

1

1’

2

2’

2

2’

1

1’

B CA D

Z

Figure 10. Scheme of discrete analysis of the field providing determination of behavior of the lines of zero amplitude. A, B, C, D – discrete plane of analysis of the field propagating along

-z axis. Filled circles depict amplitude zeroes. 1, 1’ and 2, 2’ are pairs of amplitude zeroes, which are crossings of the lines of zero amplitude with the planes A, B, C, D.

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 11. Fragments of interferograms of the field at distances m7=z m1 (a), m8=z m2

(b),

m9=z m3 (c), m=z m104

(d), ìm=z 115 (e), ìm=z 126

(f), ìm=z 137 (g), and ìm=z 148

(h)

from a rough surface. Amplitude zeroes are depicted by circles. One can observe birth (c, d) and annihilation (g) of dislocations (shown by arrows). The events of interest are shown by arrows.

(a) (b) (c) (d)

(e) (f) (g)

Figure 12. Fragments of interferograms of the field at distances ìm=z 6001 (a), ìm=z 6502

(b), ìm=z 7003 (c), ìm=z 7604

(d), ìm=z 7705 (e), ìm=z 8006

(f), and ìm=z 8107 (g)

from a rough surface. Amplitude zeroes obtained by software are depicted by circles. One can observe birth of amplitude zeroes, fragment (a), and motion of them along complex trajectory, including rotation, as the plane of observation is removed from the point of birth of singularities.

(a) (b)

(c) (d)

Figure 13. Fragments of intensity distribution (a, b) and the corresponding interferograms (c, d) at the caustics zone at distance m5=z m from an object. One can see diffraction maxima as the

lines accompanying the caustics lines, as well as relative half-period shift of interference fringes along these lines.

(a) (b)

Figure 14. Fragment of intensity distribution with the computed amplitude zeroes (shown by circles) at distance m6=z m from an object (a), and the corresponding interferogram (b).

(a) (b)

(c) (d)

Figure 15. Fragments of intensity distribution with the computed amplitude zeroes (shown by circles) at distance m6=z m from an object (a, b), and the corresponding interferograms (c, d).

(a) (b)

(c)

1

2

3

11

22

3

Figure 16. Amplitude (a) and phase (b) distributions of the field scattered by a rough surface with large inhomogeneities at distance m5=z m from the object. Fragment (c) shows the

resulting amplitude and phase distribution of the field resulting from superimposing of the distributions shown in fragments (a) and (b).

1

2

3

4

(a)

(b) (ñ)

11

22

33

44

1111

22

22

44

44

3333

Figure 17. Joint distribution of the field’s amplitude and phase (a), the same with imposed computed amplitude zeroes (depicted by circles) (b), the associated interferogram of the field (c) at distance m6=z m from the object. The areas of interest are depicted by rectangles.

(a) (b)

Figure 18. Joint distribution of the field’s amplitude and phase (a) with the computed amplitude zeroes (b) at distance ìm=z 10 from the object.

FF FF F’F’

11

22

F’F’

33

44

(a) (b)

FF F’F’

1’1’ 1’’1’’ 1’1’ 1’’1’’

2’2’ 2’’2’’

ZZ ZZ

YY

XX

00 YY00

Figure 19. To explanation of the effect of motion of the amplitude zeroes constituting a dipole. The plane FF’ place the role of the focal plane of partial focusing. The planes 1, 2, 3, 4 are the secant planes (the planes of observation). Circles depict the points of crossing of the lines of zero amplitudes with the secant planes, which are the amplitude zeroes (1’, 1’’; 2’, 2’’) at the planes of observation. Coherent wave propagates along z-axis. Fragments (a) and (b) illustrate two-dimensional and three-dimensional cases, respectively. Fragment (b) illustrates rotating and through-and-back motion of the dipole.