Huguenin et al. Vol. 20, No. 10 /October 2003 /J. Opt. Soc. Am. A 1883

Topological defects in moire fringes with spiralzone plates

J. A. O. Huguenin, B. Coutinho dos Santos, P. A. M. dos Santos, and A. Z. Khoury

Instituto de Fısica, Universidade Federal Fluminense, BR-24210-340 Niteroi, Rio de Janeiro, Brazil

Received February 12, 2003; revised manuscript received May 14, 2003; accepted May 28, 2003

We present a study of spatial structures created by superposition of spiral zone plates used for generatingoptical beams with phase singularities. Moire fringes are observed that show topological defects similar tothose appearing in interference patterns of optical vortices. A brief theoretical discussion is included thatsupports the similarities between the two phenomena. Our results may lead to interesting applications todigital information processing by optical means. © 2003 Optical Society of America

OCIS codes: 070.2580, 070.2590, 120.4120, 100.5010, 030.4070.

1. INTRODUCTIONRecently a great deal of work has been devoted to thestudy of light beams carrying phase singularities,1 alsocalled optical vortices. Both quantum and classical as-pects have been addressed in the literature. Phase-singular beams carry orbital angular momentum, whichcan be transferred to matter.2–4 Another interestingpoint regards the interaction of optical vortices with non-linear media. It has been shown that in second-harmonic generation the topological charge carried by thefundamental frequency is doubled for the second harmon-ic.5,6 Parametric downconversion with optical vorticeshas also been studied, in both spontaneous7 and stim-ulated8 cases. In the latter, it has been shown that thestimulation process follows the conservation of topologicalcharge. In the quantum domain, the orbital angular mo-mentum can be used as a further degree of freedom formanipulation of quantum information.9–12 The study ofoptical vortices has also attracted some technological re-search. The appearance of optical phase singularities inwaveguide structures has been reported,13 and high-resolution phase images of phase singularities have beenobtained in the focal length of a lens by use of an opticalfiber interferometer.14

Examples of optical vortices are the Laguerre–Gaussian (LG) modes, which are solutions of the paraxialwave equation in cylindrical coordinates.15 In thesemodes the phase depends on the azimuthal coordinate sothat a 2mp (m 5 61, 62,...) variation of the phase oc-curs in a 2p rotation about the propagation axis, where mis the topological charge. The LG modes carry also an in-teger radial index p that is related to the number of darkrings in the mode pattern. The expression for a LGmpmode is ump(r) 5 cmp(r)exp(ikz), where

cmp 5Cmp

w~z !S rA2

w~z !D umu

LpumuF 2r2

w2~z !GexpF2

r2

w2~z !G

3 expH iF kr2

2R~z !2 ~2p 1 umu 1 1 !tan21S z

zRD

1 mfG J . (1)

1084-7529/2003/101883-07$15.00 ©

Here Cmp is a normalization constant, k is the wave num-ber, w(z) is the beam radius, Lp

m is the associated La-guerre polynomial, R(z) is the wave-front radius, and zRis the Rayleigh length. Optical vortices are also con-ceived beyond the paraxial approximation. Exact solu-tions of the Helmholtz equation that possess phase singu-larities have been discussed in Refs. 16–18. In thesereferences it is described how the LG modes [Eq. (1)] arisefrom the paraxial approximation in more-general solu-tions given by the so-called Bessel beams.

The available methods for generating optical vorticesare based essentially on either holographic methods19,20

or astigmatic mode conversion.21,22 In the latter, aTEM01 Hermite–Gaussian mode is incident on a pair ofcylindrical lenses suitably placed so as to transform theinput mode into an LG vortex mode. The holographicmethod has some advantages because of its simplicity,whereas astigmatic mode converters allow for low lossesand are more adequate when power is essential. The ho-lograms used for vortex generation are a variation of thewell-known Fresnel zone plates (FZPs) [Fig. 1(a)] that arefrequently used as either an amplitude or a phase maskfor focusing. In addition to the usual focusing effect, spi-ral zone plates (SZPs) such as those shown in Figs. 1(b)and 1(c) can generate optical vortices. The number andthe sense of the dark (or the white) spirals determine thetopological charge of the vortex.

It is well known that superposition of two spatially re-petitive structures (not necessarily periodic) gives rise tonew structures that are sometimes not contained in ei-ther of the superposed ones, leading to the so-called moirefringes.23 Moire patterns are traditionally employed inoptical metrology, for example, as a nondestructive test-ing routine in mechanical engineering.24–26 Optical ap-plications based on moire like patterns have been ex-tended to many different fields. As an example, we canmention speckle shear interferometry techniques,27 laserproperty measurements,28 microscopy,29 and advanced li-thography processes in microeletronics. Moire fringesare well known for a variety of structures including theFZP, with which they appear as a sequence of parallelstraight lines. In this paper we study the moire fringes

2003 Optical Society of America

1884 J. Opt. Soc. Am. A/Vol. 20, No. 10 /October 2003 Huguenin et al.

Fig. 1. Zone plates for (a) m 5 0 (FZP), (b) m 5 1, and (c) m 5 2.

generated by superposition of two SZPs. Topological de-fects were observed that were similar to the vortex signa-ture in interference fringes. We also present a theoreti-cal treatment based on that in Ref. 30, which supports ourexperimental results. These results suggest interestingapplications to information processing, including cryptog-raphy, as we shall discuss in our conclusion.

2. FOURIER DECOMPOSITIONThe notion of a repetitive structure is well described inRef. 30. Such structures are not strictly periodic butpresent some kind of repetition governed by mathemati-cal rules. The SPZs studied here belongs to a class ofsuch repetitive structures, namely, the coordinate trans-formed structures. The transmission function associatedwith such structures can be written as t(x, y)5 p@ g(x, y)#, where p(x8) is some periodic functioncalled the periodic profile and x8 5 g(x, y) is a coordinatetransformation g : R2°R that gives the geometric layoutof the structure. For example, a cosinusoidal FZP can beproduced by combining p(x8) 5 cos(2pnx8) with g(x, y)5 a(x2 1 y2). In general, the periodic layout p(x) canbe any periodic waveform. A binary structure (black/white), for example, corresponds to a square wave. TheSPZs are described by the transformation g(x, y)5 a(x2 1 y2) 1 m arg(x 1 iy) or, in polar coordinates,g(r, f ) 5 ar2 1 mf. The case m 5 0 clearly gives theFZP.

The spectral properties of the transmission function isdetermined by the Fourier decomposition of the periodicprofile p(x8):

p~x8! 5 (n52`

`

cn exp~2ipnnx8!. (2)

For an amplitude mask (gray scale) the transmissionfunction is real and the spatial Fourier spectrum containspositive and negative frequencies with equal weights. Inthis paper we shall limit our analysis to amplitude masksand leave the study of phase masks to the future. Thepresence of positive and negative frequencies gives rise tospatial beat notes when two amplitude masks are super-posed. This is in the heart of the moire effect leading tointeresting structures when SZPs are employed.

3. SUPERPOSITION MOIRES BETWEENSPIRAL ZONE PLATESWhen two structures are superposed, each layer is repre-sented by a transmittance function ti(r, f ), assumingvalues from 0 (black) to 1 (white). In our case it will bemore convenient to express the transmittance function inpolar coordinates. The superposition is then describedby the product of the individual layers:

t~r, f ! 5 t1~r, f !t2~r, f !. (3)

The SZP belongs to the class of coordinate transformedstructures for which the transmittance functions can bewritten as ti(r, f ) 5 p(x8), where p is some periodicfunction and x8 5 gi(r, f ) represents the coordinatetransformation. A general expression for the zone platesstudied here is

g~r, f ! 5 ar2 1 mf, (4)

where a gives the radial scale and m is the angular orderthat determines the topological charge for vortex genera-tion. The value m 5 0 reproduces the usual Fresnelzone plate. By taking the Fourier expansion for p(x8)and writing the transmittance functions as

ti~r, f ! 5 (n52`

`

cn~i ! exp@2ipngi~r, f !#, (5)

we obtain the Fourier expansion for the superposed struc-tures:

t1~r, f !t2~r, f !

5 (n,l52`

`

cn~1 !cl

~2 ! exp$2ip@ng1~r, f ! 1 lg2~r, f !#%. (6)

The moire fringes can be extracted from the expansionabove as the partial sums:

fk1 ,k2~r, f ! 5 (

m52`

`

cmk1

~1 ! cmk2

~2 ! exp@2ipmgk1 ,k2~r, f !#,

(7)

where k1 and k2 are co-prime integers and gk1 ,k2(r, f )

5 k1g1(r, f ) 1 k2g2(r, f ) is the geometric layout ofthe moire structure. The structure fk1 ,k2

is not presentin any of the original ones but appears in the superposi-tion pattern. Usually, only the lowest-order moires fall

Huguenin et al. Vol. 20, No. 10 /October 2003 /J. Opt. Soc. Am. A 1885

in the visible frequency range; therefore we shall focus onthe f1,21 contribution. Let us first consider the superpo-sition of two SZPs with topological charges m1 and m2 ,slightly displaced along x such that

g1~r, f ! 5 a@~x 1 e/2!2 1 y2# 1 m1f,

g2~r, f ! 5 a@~x 2 e/2!2 1 y2# 1 m2f, (8)

with e ! 1. Therefore, keeping terms up to first order ine, we obtain

g1,21~r, f ! 5 2aex 1 Dmf

5 2aer cos f 1 Dmf, (9)

where Dm 5 m1 2 m2 . The resulting geometric layoutis a set of curves representing dark and white fringes:g1,21(r, f ) 5 qp (q 5 0, 61,...). For the usual case ofFZP (m1 5 m2 5 0), one obtains a sequence of straightlines 2aex 5 qp, regularly spaced. The same layout isobtained when two SZPs with m1 5 m2 are superposed.For m1 Þ m2 , g1,21 becomes dependent on f, which is ill-defined at the origin. Therefore a topological defect ap-pears on the geometric layout of the moire fringes. Inpractical terms, Dm bifurcations appear in the geometriclayout, in analogy with the interference patterns of opti-cal vortices.

Another interesting structure arises when two SZPswith different radial scales are superposed, so that

g1~r, f ! 5 a1r2 1 m1f,(10)

g2~r, f ! 5 a2r2 1 m2f.

In this case we obtain

g1,21~r, f ! 5 Dar2 1 Dmf, (11)

which means that the resulting structure is again an SZPwith topological charge Dm 5 m1 2 m2 and radial scaleDa 5 a1 2 a2 .

The relation Dm 5 m1 2 m2 following both Eqs. (9)and (11) are analogous to the indicial equations appearingin Ref. 24 [Sec. 1.4, Eq. (1.9)]. In fact, there is a comple-mentarity between Eq. (9) and the one presented in Ref.24. In the latter, the corresponding indicial equation re-fers to the superposition of two rotated straight-linestructures, while Eq. (9) refers to the superposition of twodisplaced curved gratings.

4. EXPERIMENTS AND DISCUSSIONTo illustrate the similarities between moire fringes andinterference patterns, we first performed interferometricmeasurements with the setups shown in Fig. 2. One ofthe outputs of a Mach–Zender interferometer is regis-tered with a CCD camera. Optical vortices are generatedby sending a laser beam to holograms like those in Fig. 1.As already mentioned, the holograms work like a zoneplate, focusing the beam to a focal length fH . By place-ment of a lens (L1, focal length fL) after the hologram in aconfocal configuration, a collimated optical vortex is ob-tained. Depending on the desired interference, the opti-cal vortex is generated outside [Fig. 2(a)] or inside [Fig.2(b)] the interferometer.

In the right column of Fig. 3 we present the moirefringes obtained by superposition of two SZPs. One ofthem was placed on an X –Y stage with micrometer con-trol to produce small displacements e along the X direc-tion. The superposed SZPs were illuminated by whitelight, and the resulting image with moire fringes was cap-tured by a CCD camera. In the left column the analo-gous interference patterns between optical vortices areshown. In Fig. 3(a) we present the self-interference pat-tern for an m 5 1 optical vortex, produced by diffractionof a 7-mW He–Ne laser beam (l 5 632.8 nm) by an SZPwith m 5 1. The optical vortex traverses a Mach–Zender interferometer whose output was captured by aCCD camera [Fig. 2(a)]. The interferometer is slightlymisaligned in order to provide a spatial interference pat-tern. The corresponding moire fringes between twom 5 1 masks are shown in Fig. 3(d). The similarity be-tween the interference pattern and the correspondingmoire fringes is very clear. In both cases straight linesappear.

For the result presented in Fig. 3(b), an m 5 0 beamwas sent into the interferometer and the optical vortexwas produced in one arm [Fig. 2(b)], thus providing theinterference between m 5 1 and m 5 0. On the otherhand, in Fig. 3(e) the moire pattern resulting from the su-perposition of an FZP with m 5 0 and an SZP with m5 1 is shown. Again, the same topological defect arisesin both cases, corresponding to Dm 5 1. However, acurved bifurcation appears in Fig. 3(e), which is due to asmall difference in the radial scale between the two zoneplates, as we shall describe shortly.

In Fig. 3(c) the interference pattern between m 5 1and m 5 21 optical vortices is shown. To provide them 5 21 beam, a Dove prism (DP) was inserted into onearm of the interferometer [Fig. 2(a) with DP]. The analo-gous moire fringes are shown in Fig. 3(f ); a topological de-fect corresponding to Dm 5 2 is clear in both cases.

Fig. 2. Experimental setup for interferometric measurements.H, hologram; L, lens; M, mirror; BS, beam splitter; DP, Doveprism, D 5 fH 1 fL . (a) Setup used for self-interference of anm 5 1 optical vortex and for interference between m 5 1 andm 5 21. (b) setup used for interference between m 5 0 andm 5 1.

1886 J. Opt. Soc. Am. A/Vol. 20, No. 10 /October 2003 Huguenin et al.

Fig. 3. Interference patterns between two optical vortices with (a) m1 5 m2 5 1, (b) m1 5 1 and m2 5 0, (c) m1 5 1 and m2 5 21, andanalogous moire fringes for superposition of two spiral zone plates: (d) m1 5 m2 5 1, (e) m1 5 1 and m2 5 0, (f ) m1 5 1 and m25 21.

In order to produce a given hologram, the desired lay-out was computer generated and photographed with ahigh-resolution film (6 ASA) to provide a gray-scale mask.This procedure allowed the production of nearly perfectcopies in a single run but could not guarantee preciselyequal radial scales for two different runs. Therefore, formeasurements with two zone plates having the same ab-solute value of m, two copies of the same run were used.For m1 5 2m2 , for example, two nearly identical maskswere used and one of them was just flipped with respectto the other. On the other hand, for m1 5 1 and m25 0 the two masks were obtained in different runs andthe radial scale was determined with limited accuracy.Since the moire pattern amplifies these imperfections, acurved bifurcation appears.

This effect motivated us to investigate the moire pat-terns resulting from superposition of SZPs with different

radial scales, as predicted by Eq. (11). An interferencepattern analogy applies also to this case. Actually, theinterference between two divergent optical vortices givesrise to the same spiral fringes. In Fig. 4 we present themoire fringes obtained from superposition of two SZPswith different radial scales, together with the correspond-ing interference patterns. The low-frequency spiral lay-out is clear in the moire fringes. In Fig. 4(a) the self-interference of an m 5 1 optical vortex is shown. Asbefore, an m 5 1 optical vortex is produced by diffractionof a laser beam by an SZP with m 5 1. This vortex issent to a Mach–Zender interferometer, which is now wellaligned. A lens (L2) is then introduced into one arm ofthe interferometer to provide a divergent beam [Fig. 2(a)with L2 and without DP]. The resulting interferencepattern shows the usual Fresnel ring structure. In Fig.4(d) the corresponding moire structure is presented,

Huguenin et al. Vol. 20, No. 10 /October 2003 /J. Opt. Soc. Am. A 1887

which appears from the superposition of two SZPs withm 5 1 that have different radial scales. The first ring ofa low-frequency Fresnel structure (Dm 5 0) can be seeneasily.

In Fig. 4(b) the interference between an optical vortexwith m 5 1 and a divergent TEM00 (m 5 0) beam isshown. The setup used is the one in Fig. 2(b) with L2 in-serted. Spiral fringes are evident; they can also be seenin Fig. 4(e), where the moire pattern produced by super-position of an FZP and an SZP with m 5 1 is shown.The moire structure corresponding to Dm 5 1 appears asa low-frequency dark spiral. The situation is similar inFig. 4(c), where the interference pattern between a diver-gent m 5 1 optical vortex and a collimated m 5 21

beam is shown [setup in Fig. 2(a) with L2 and DP in-serted]. A double spiral can be seen corresponding toDm 5 2. The same structure is evident in Fig. 4(f ),where the moire pattern from the superposition of an m5 1 and an m 5 21 SZP is shown. Again a low-frequency double spiral can be seen clearly.

To illustrate the moire fringes from higher-order struc-tures, in Fig. 5 we present the moire patterns generatedby superposition of SZPs with umu 5 2. The SZPs usedhere were produced at large scale to provide a better reso-lution, thus allowing the appearance of many moirefringes. In Figs. 3 and 4, on the other hand, we employedthe same SZP both for vortex generation and for moirepattern formation, since in that case we were comparing

Fig. 4. Interference patterns between a divergent optical vortex and a collimated one for (a) m1 5 m2 5 1, (b) m1 5 1 and m2 5 0, (c)m1 5 1 and m2 5 21, and analogous moire fringes for spiral zone plates with different radial scales: (d) m1 5 m2 5 1, (e) m1 5 1 andm2 5 0, (f ) m1 5 1 and m2 5 21.

1888 J. Opt. Soc. Am. A/Vol. 20, No. 10 /October 2003 Huguenin et al.

Fig. 5. Moire fringes between spiral zone plates with (a) m1 5 m2 5 2, (b) m1 5 2 and m2 5 22, and moire fringes for spiral zoneplates with different radial scales: (c) m1 5 m2 5 2, (d) m1 5 2 and m2 5 22.

the two phenomena. In Figs. 5(a) and 5(b), two SZPswith the same radial scale were superposed and slightlydisplaced along the horizontal direction. The straight-line structure is clear for m1 5 m2 5 2, whereas a mul-tiple bifurcation appears when m1 5 2m2 5 2. In Figs.5(c) and 5(d) two SZPs with different radial scales are su-perposed. The first rings of the Fresnel structure can beseen clearly from Fig. 5(c), where we used m15 m2 5 2, while four low-frequency dark (or white) spi-rals are evident from Fig. 5(d). In all cases the struc-tures observed are consistent with the values of Dm5 m1 2 m2 as given by Eqs. (9) and (11).

One may envisage, for example, an interesting applica-tion of the moire effect with SZP to cryptography. Let ussuppose that some digital information, composed of a se-quence of integer numbers, is encoded on a set of holo-grams according to some secret rule (the addition of agiven random sequence, for example). The coded infor-mation would then be another sequence of integer valuescorresponding to the topological charges in the hologramset. To recover the original information, one can use asecond hologram set, the reading one, such that superpo-sition of the reading set and the coded one would invertthe coding rule, yielding the original information throughthe resulting moire structures. The original informationis therefore given by the topological charges in the moiresequence, which can be read by optical means.

5. CONCLUSIONSIn summary, we have demonstrated the appearance of to-pological defects in moire fringes obtained from superpo-

sition of SZPs. These defects are analogous to those thatare typical of interference patterns with optical vortices.Our experimental results are supported by a Fourier-based analysis of moires in the superposition of geometri-cally transformed periodic structures. In fact, the SZPsstudied here are a special class of such structures. Themathematics that govern the topological charge of themoire structure in terms of the original ones may lead tointeresting applications to digital information processingby optical means.

ACKNOWLEDGMENTSThe authors acknowledge financial support from the Bra-zilian agencies Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico (CNPq), Coordenacao de Aper-feicoamento de Pessoal de Nıvel Superior (CAPES), andFundacao de Amparo a Pesquisa do Estado do Rio de Ja-neiro (FAPERJ).

Corresponding author Antonio Khoury’s e-mail addressis khoury@if.uff.br.

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