Feasibilities of interferometric and chromascopic techniques in study of phase singularities

Feasibilities of interferometric and chromascopictechniques in study of phase singularities

Oleg V. Angelsky, Steen G. Hanson, Alexander P. Maksimyak, and Peter P. Maksimyak

The feasibilities of using interferometric and chromascopic techniques in the diagnostics of phase sin-gularities and in the study of a phase structure of the field in their vicinity are demonstrated. The peculiarevolution of singularities into caustics produced by phase elements of singularity-generating objects ofspherical and cylindrical shape is studied. © 2005 Optical Society of America

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

1. Introduction

Investigation of the field structure in the vicinity ofsingular points (amplitude zeros, vortices, or wave-front dislocations) is of importance both in funda-mental optics and associated applications.1–4 Inves-tigation, on the one hand, leads to new knowledge ofthe field structure and, on the other hand, providesnew potentials for the use of optical techniques innanotechnologies, where submicron wave structuresare the main instrument.5 So, in part, phase struc-ture of the field in the vicinity of amplitude zerocontains the data on topology of a wave dislocation,topological charge, and degree of asymmetry.

In this context, the study of caustics in the field ofdiffracted (or scattered) radiation attracts growingattention. Caustics occur as the main feature in theformation of singularities in the field, including com-plex speckle fields.6

Our purpose in this paper is to conduct computersimulations and experimental verification of the sce-nario of evolution and peculiarities of singularitiesproduced by phase elements of singular-generatingobjects of cylindrical and spherical shapes using in-terferometric and chromascopic techniques.

2. Interferometric Study of Phase Singularities

As the first step, we study the field structure using aninterferometric technique.7 The essence of this tech-nique consists of analysis of the interference distri-bution resulting from superposition of the objectbeam with the coherent reference beam. Spacing andorientation of interference fringes may be controlledboth by simulation and experimentally through vary-ing the interference angle. All the above-mentionedprovide:

Y observation of amplitude and phase structureof the field at various scales, including fragmentswith a size of several wavelengths;

Y precise interferometric determination of theloci of amplitude zeros defined by the loci of typicalinterference forklets (bifurcations of interferencefringes);

Y interferometric exploration (with variable res-olution determined by the orientation of the referencewave) of the fragments of a field, whose phase struc-ture at various cross sections is of interest.

To calculate the field transmitted by a cylindricallens, we use the one-dimensional Rayleigh–Sommerfeld diffraction integral that provides compu-tation of the field at an arbitrary observation plane,both in front of and behind a lens8:

U(�) �z

�i�� F(x)

R3�2(x)exp{�ik[R(x) � h(x)]}dx, (1)

where F�x� is the aperture function of a lens, R�x�� z � h�x��2 � �x � ��21�2 is the distance from anobject point to the point of observation, h�x� is theprofile of the cylindrical lens, z is the distance from

O. V. Angelsky, A. P. Maksimyak, and P. P. Maksimyak are withthe Department of Correlation Optics, Chernivtsi University, 2Kotsyubinsky Street, Chernivtsi 58012, Ukraine. S. G. Hanson([email protected]) is with the Department of Optics andFluid Dynamics, RISO National Laboratory, P.O. Box 49, RoskildeDK-4000, Denmark.

Received 22 November 2004; revised manuscript received 15February 2005; accepted 17 February 2005.

0003-6935/05/245091-10$15.00/0© 2005 Optical Society of America

20 August 2005 � Vol. 44, No. 24 � APPLIED OPTICS 5091

the object to the registration plane, and x, � are rect-angular Cartesian coordinates in the object plane andin the plane of observation, respectively, cf. Fig. 1.

We perform calculations for various distances of theboundary “glass–air.” In our study, we change theshape of this boundary (from cylindrical with variousradii of curvature and orientation to spherical). Wealso vary the intensity distribution at the cross sectionof the probing beam from uniform to Gaussian, keep-ing the wavefront of the probing beam plane. Further,we use the previously introduced interference tech-nique for studying the phase structure of the field atvarious registration zones.9 This facilitates the obser-vation of the interference pattern, from which one canconclude phase and amplitude distribution of the ob-ject field U��. These results are compared with thedirectly computed phase distributions of the field.

Let us consider Fig. 2, where the intensity and thephase distributions of the fields passing a cylindricallens and reflected from its output surface are shown.The general structure of these distributions has pre-viously been demonstrated.10,11

One can see a typical pyramidal structure with atriangular cross section decorated with outer lightbeams. The field distribution inside the triangle is ofquasi-periodic structure, where one can distinguishthe axial direction and the direction parallel to thesides of the triangle. Comparing the intensity andphase distributions represented at the same scale[Figs. 2(a) and 2(b)], one can precisely determine thefocal plane by two characteristics. Namely, this planecorresponds to the maximal intensity at the focal spotadjoining the triangle vertex [depicted in Fig. 2(a)],coinciding with the locus of quasi-plane fragment of awavefront [depicted in Fig. 2(b)], at the boundary ofwhich one observes the bifurcations of the planes ofconstant phase.

The interference distribution obtained as a resultof coherent superposition of the field of interest witha plane reference wave [Fig. 2(c)] possesses the typ-ical interference forklets at the focal plane associatedwith the phase singularities. So, we interferometri-cally visualize an amplitude and phase distributioncorresponding to the Fraunhofer slit diffraction [Fig.2(d), enlarged segment c3].

More detailed analysis of an interference pattern

shows the presence of phase singularities (amplitudezeros) at each of the “dark” zones between the beams,which are thought to emanate from the triangle [Fig.2(d), segment c2].

Inside of the triangle [Fig. 2(d); b2 and c2] one alsoobserves periodically distributed amplitude zeros, incorrespondence with the concept of the Pearcey dif-fraction pattern.12

The same pattern (though with less spacing of am-plitude zeros) is observed at the radiation field re-flected from the cylindrical surface of a lens [Fig. 2(d);a1, b1, and c1].

To compute the field passing a spherical lens, weuse a two-dimensional Rayleigh–Sommerfeld diffrac-tion integral. The intensity and phase distributions ofthe field obtained for this case (Fig. 3) do not differsignificantly from the distributions for the field pass-ing a cylindrical lens.

The evolution of singularities for the field as a func-tion of the numerical aperture of a lens (ratio of theaperture of a lens a to the focus length f of a cylin-drical surface) can be obtained from the analysis ofamplitude and phase distributions of the field shownin Fig. 4 for a cylindrical lens with a radius of curva-ture of 10 �m for apertures a�f: 0.5 (a), 0.65 (b), and0.75 (c). The distributions for the unity aperture of alens with a curvature radius 20 �m (for refractiveindex 1.5) are represented in Figs. 2(a) and 2(b).

Note that the peculiarities of this scenario diag-nosed interferometrically are in good agreement withthe statements in a previous paper.10,11 In fact, oneobserves passing from a planar Fraunhofer diffrac-tion pattern [Fig. 4(a)] to the volume Pearcey patternas the aperture of the cylindrical lens increases [Figs.2(a) and 2(b)]. At the same time, it is worth notingthat the amplitude zeros inside the triangle initiallynucleate at zones neighboring the surface of the cy-lindrical lens [Fig. 4(b)]. We calculate the fields forcylinders with various radii and for various aper-tures. For cylinders of various radii R, the aperturesa�f for which the phase singularities disappear aredifferent also, as is seen from Fig. 5. It is impossibleto explain the presented dependencies, characteriz-ing the mechanisms of nucleation and disappearanceof the phase singularities inside of the triangle, tak-ing into account lens aberrations only. Moreover, thecalculation performed for a tautochronic system (i.e.,for a system obeying the generalized Fermat’s prin-ciple), which is an aberration-free system, gives asimilar structure for the fields (Fig. 6). One can ob-serve some changes in the amplitude and phase dis-tributions of the field within the triangle. Continuousplanes of constant amplitude are noticed. But as awhole, the outline of the distribution does not change.

One can imagine that this effect is associated withthe sharp aperture limitations of the beam, so thatthe so-called boundary diffraction wave13–15 is in-volved in the formation of the field inside the triangle.This hypothesis is confirmed by the fact that thephase singularities within the triangle disappearwhen the probing beam has a Gaussian amplitudeenvelope, as is seen from Fig. 7.

Fig. 1. Formation of the field diffracted by a cylindrical lens.

5092 APPLIED OPTICS � Vol. 44, No. 24 � 20 August 2005

Thus, one can assert that a cylindrical lens is thesingularity-generating object, if the amplitude of theprobing beam at the lens rim is sufficient for produc-ing a boundary wave of considerable magnitude. Ithas been stated that for a sufficiently large apertureof a cylindrical lens, the triangle limiting the converg-ing beams is filled with phase singularities, whosedistribution is spatially well determined. Such struc-ture can be diagnosed interferometrically, at hasbeen confirmed by computer simulation. The possi-bility of experimental diagnostics of complex specklefields has been demonstrated recently,6 where it wasbeen stated that the scenario of nucleation of phasesingularities is created by the focal zone and the ar-

eas adjacent to the cylinder surface. It has beenshown that the use of a Gaussian probing beam re-sults in the disappearance of the phase singularitiesinside the triangle of the focused beam.

3. Exploring Phase Singularities Using a Chromascope

The above consideration shows the high efficiency ofthe interferometric study of phase singularities in thefield using a coherent reference wave. However, theexperimental implementation of the interferometrictechnique is often hampered or even impossible inproblems including micro- and nanotechnologies.That is why one must look for another experimentalmeans for the study of the field structure in the vi-

Fig. 2. (a) Intensity and (b) phase distributions of the fields passing a cylindrical lens; (c) interference distribution obtained as a resultof superposition with a plane reference wave; (d) enlarged segments depicted in (a), (b), and (c).


cinity of the amplitude zeros. Recent results concern-ing strong spectral distortions near the zeros ofdiffraction patterns formed in polychromaticlight16–20 seem at this point to be a promising tool.Here we use the concept of a chromascope introducedin Ref. 16. The essence of this concept consists innormalizing the colors that provide clear visualiza-tion of colors within the areas of low illumination(dark regions) near the amplitude zeros.

We simulate illumination of a cylindrical lens by aplane wave consisting of three spectral components,with wavelengths 633 nm, 540 nm, and 430 nm, con-stituting a red-green-blue (RGB) set. To reveal thecolors, the RGB values at each point are scaled toisoluminance by the transformation

RGB�)R

GB��max(R, G, B). (2)

This “chromascope” preserves the ratios between thethree RGB values while making the biggest one equalto unity, so the corresponding point of the image is asbright as possible. It should be emphasized, however,

Fig. 3. (a) Intensity and (b) phase distributions of the field ob-tained for a spherical lens.

Fig. 4. Intensity (left) and phase (right) distributions of the fieldobtained for cylindrical lens with a radius of curvature of 10 �m forapertures a�f: (a) 0.5, (b) 0.65, and (c) 0.75.

Fig. 5. Dependence of a�f ratio on the radius of cylindrical lens R,for which the phase singularity disappears.


that the scaling expression (2) is a definition, appro-priate for the present purpose of revealing colors inthe dark part of the images.

The use of independent colors provides a reliablesolution to the diagnostic problem using the addi-tional equations related to each of the colors, whichconnect the parameters of the diagnosed object andits scattered field.

Using the Rayleigh–Sommerfeld diffraction inte-gral Eq. (1), we compute the field for each spectralcomponent and add the intensities for the spatialdistributions. The result of calculation of the summedover colors field intensities is shown in Fig. 8(a) forthe beam passing a cylindrical lens with the radius ofcurvature of a surface 10 �m. The use of a chromas-cope provides the pattern shown in Fig. 8(b).

We used three colors simultaneously and consid-ered their partial combinations. In case of coincidingamplitude zeros for all three color components, we

use the chromascope results in the pattern shown inFig. 9(a). In the point of zero amplitude, all colorsmeet. As a result, one observes a radial structurewhere the space between two generating lines is filledby a single color before and behind the point of inter-ception. Figure 9(b) illustrates the case when theamplitude zeros for two colors coincide.

Three fragments of the patterns similar in struc-ture to the ones processed by a chromascope for var-ious scales for a cylindrical lens with a curvatureradius 100 �m at the distances (a) 10 �m, (b) 30 �m,and (c) 110 �m from the apex point of the cylindricallens are shown in Fig. 10. Analysis of the representedcolor distributions shows quasi periodicity of the par-tial color combinations, which are depicted in thisfigure. The longitudinal and transverse scales of thedistribution are proportional, respectively, to �f�a�2

and �f�a�, a being the size of a lens aperture, and fbeing the focal length.

Fig. 6. (a) Intensity and (b) phase distributions of the field for atautochronic lens with a radius of curvature of 10 �m.

Fig. 7. (a) Intensity and (b) phase distributions of the field for acylindrical lens illuminated with a Gaussian beam.


Rough surfaces are more complex singularity-generating objects involving quasi-spherical andquasi-cylindrical elements and elements of higher cur-vatures. Scattering of coherent radiation at such arough surface results in the formation of a fully devel-oped speckle field with numerous amplitude zeros.That is why we have investigated the phase singulari-ties for a field scattered by a rough surface. We simu-lated plane-wave diffraction for a phase reliefmimicking a rough surface. The height of inhomoge-neities of a surface (N � N pixels) is specified for eachpixel by the Gaussian random-number generator, fol-lowed by a two-dimensional smoothing process withspecified dispersion. Figure 11 illustrates an exampleof a surface (300 � 300 pixels) smoothed over threepixels.

The intensity and phase distributions of the field

resulting from the diffraction of a plane wave from arough surface are computed using the doubleRayleigh–Sommerfeld diffraction integral. Figure 12shows (a) the intensity distribution and (b) the phasedistribution of the diffracted field, as well as (c) thepattern processed by a chromascope for a radiationconsisting of the components with three wavelengths.In Fig. 12(a) one can see the minimum for all threespectral components at the center of the pattern. InFig. 12(b) one observes a break in the equiphase linesand a following spatial blurring for each spectral com-ponent. This confirms the colocation of the amplitudezeros at the center of the pattern. The use of a chro-mascope [Fig. 12(c)] determines the location of theamplitude zero as the point where several colorsmeet.

Figure 13 illustrates the result of diffraction of aplane wave from a rough surface in case two wave-

Fig. 8. (a) Intensity distributions of the field for cylindrical lenswith a radius of curvature of 10 �m illuminated with a beamconsisting of three spectral components. (b) The use of a chromo-scope provides the pattern.

Fig. 9. (a) Patterns obtained by the use of a chromascope forcoinciding amplitude zeros of three spectral components, (b) fortwo spectral components.


lengths are used. Amplitude zeros for both compo-nents coincide in the center of the pattern.

The reported results show that using a chromas-cope one can determine the loci of amplitude zeros ofthe field in case of colocation of the amplitude zerosfor distinct spectral components.

Fig. 12. (a) Intensity and (b) phase distributions of the field dif-fracted by a rough surface, and the pattern resulting from (c)chromascopic processing for a polychromatic field consisting ofthree wavelengths. The amplitude zeros for three spectral compo-nents coincide in the center of the pattern.

Fig. 10. Patterns processed by a chromascope for various scalesfor a cylindrical lens with a curvature radius 100 �m at the dis-tances (a) 10 �m, (b) 30 �m, and (c) 110 �m from the apex point ofthe cylindrical lens.

Fig. 11. Relief map for the modeled surface with a three-pointsmoothed random surface obeying a Gaussian height distribution.


4. Experimental Study of Singularities at the BeamsPassing Double-Axial Crystal and Ground Glass

To experimentally prove the benefits of the consid-ered technique, we use a polyethylenterephthallium(PETP) film possessing the properties of a double-axial crystal (thickness 74 �m, difference in refrac-tion indices 0.085). To form an optical singularity, weuse the optical arrangement shown in Fig. 14.

An objective 2 images a white-light source 1 into a50-�m-diameter diaphragm 3, which together withthe objective 4 (focal length 200 mm) forms a beamwith a high degree of spatial coherence � 95%�. Oneof the optical axes of the crystal coincides with the

Fig. 14. Experimental optical arrangement: 1, source of light; 2,4, 8, objectives; 3, diaphragm; 5, polarizer; 6, film; 7, analyzer; 9,CCD camera; 10, computer.

Fig. 15. Conoscopic patterns obtained in white light for (a) par-allel and (b) crossed polarizer and analyzer.

Fig. 13. (a) Intensity and (b) phase distributions of the field dif-fracted by a rough surface, and the pattern resulting from (c) achromascopic processing for a polychromatic field consisting ofthree wavelengths. Amplitude zeros for two spectral componentscoincide in the center of the pattern.


optical axis of the beam. An objective 4 images thediaphragm 3 into the film 6. A polarizer 5 and ananalyzer 7 are rotated with respect to the optical axisand provide observation both with collinear andcrossed axes of transmission. An objective 8 imagesthe film 6 onto the sensitive area of the CCD camera9.

The conoscopic patterns obtained in white light fortwo orientations of a polarizer are shown in Fig. 15.In Fig. 15(a) one observes an amplitude zero (for allspectral components) as is seen from the distributionof colors and their sequence.20 The use of a chroma-scope provides means for obtaining the classical pat-tern for such a singularity (Fig. 16). The feasibilitiesfor conversion of singular beams (however, using uni-axial crystals) have been shown.21

Observation of the polychromatic radiation fieldscattered by a rough surface with phase variance ofirregularities 20 rad using the same optical ar-rangement shows the presence of color speckles in theFresnel zone with respect to the average irregularityof the surface (Fig. 17). The use of a chromascope forprocessing the experimentally obtained patterns pro-vides diagnostics of polychromatic amplitude zerosfor specified components of the field scattered by arough surface (Fig. 18). The difference in the color

pattern obtained experimentally and the amplitudezero of the field resulting from a computer simulation(in experiments both with a rough surface and with amonocrystalline film) is caused by the fact that theexperiment has been performed with a white-lightradiation source (a halogen lamp), while the com-puter simulation is based on a triple-wavelengthsource.

Note that application of a chromascope for experi-mental diagnostics of the white-light vortex had beenrecently demonstrated,22 devoted to synthesis of thewhite-light singularities with a different value of to-pological charge.

5. Conclusion

We have discussed the feasibilities of a chromascopictechnique for determination of the loci of phase sin-gularities and for the analysis of the field structure inthe vicinity of amplitude zeros. The obtained resultsprovide, in our opinion, a promising tool for the prob-lem of looking for the amplitude zeros in polychro-matic radiation. Here, it is important to distinguishthe points where two or three RGB components of a

Fig. 16. Pattern shown in Fig. 14 processed by a chromascope.

Fig. 17. Polychromatic radiation field scattered by a rough sur-face.

Fig. 18. Pattern obtained by applying a chromascope to the ex-perimentally found intensity distributions.


field vanish simultaneously, to determine the chargeof phase singularities, as well as to distinguish anamplitude zero from amplitude minima without astrict amplitude zero. These are problems for ourfurther investigation.

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Feasibilities of interferometric and chromascopic techniques in study of phase singularities