Airy pattern reorganization and subwavelength structure in ...web.physics.ucsb.edu/~quopt/airy.pdf · Airy pattern reorganization and subwavelength structure in a focus G. P. Karman,

884 J. Opt. Soc. Am. A/Vol. 15, No. 4 /April 1998 Karman et al.

Airy pattern reorganization and subwavelengthstructure in a focus

G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman

Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands

Received July 21, 1997; revised manuscript received October 15, 1997; accepted October 20, 1997

An early result of optical focusing theory is the Lommel field, resulting from a uniformly illuminated lens; thedark rings in the focal plane, the Airy rings, have been recognized as phase singularities. On the other hand,it is well known that Gaussian illumination leads to a Gaussian beam in the focal region without phase sin-gularities. We report a theoretical and experimental study of the transition between the two cases. Theo-retically, we studied this transition both within and outside the paraxial limit by means of diffraction theory.We show that in the gradual transition from uniform toward Gaussian illumination, the Airy rings reorganizethemselves by means of a creation/annihilation process of the singularities. The most pronounced effect is theoccurrence of extra dark rings (phase singularities) in front of and behind the focal plane. We demonstratetheoretically that one can bring these rings arbitrarily close together, thus leading to structures on a scalearbitrarily smaller than 1 wavelength, although at low intensities. Experimentally, we have studied the con-sequences of the reorganization process in the paraxial limit at optical wavelengths. To this end, we devel-oped a technique to measure the three-dimensional intensity (3D) distribution of a focal field. We applied thistechnique in the study of truncated Gaussian beams; the experimentally obtained 3D intensity distributionsconfirm the existence and the reorganization of extra dark rings outside the focal plane. © 1998 Optical So-ciety of America [S0740-3232(98)01603-2]

OCIS codes: 050.1940, 220.2560.

1. INTRODUCTIONThe focal field produced by a lens has been extensivelystudied in the past by many researchers. A very early re-sult is the so-called Lommel field, which is the focal fieldthat results when one uniformly illuminates the lens inthe paraxial limit. In this limit one can analyticallysolve the Huygens–Fresnel diffraction integral in termsof so-called Lommel functions.1 A prominent feature ofthe Lommel field is the Airy pattern in the focal plane: apattern of concentric dark and bright rings. In the pastthese dark rings have been recognized as phasesingularities.2,3 The concept of phase singularities inwave fields was emphasized by Nye and Berry4,5; phasesingularities are defined as points in space where the gra-dient of the phase diverges and where the phase itself isundefined. A consequence is that at such a point the am-plitude of the wave field is identically zero. Another ex-ample of a phase singularity in optics is the axis of aLaguerre–Gaussian donut beam. As Nye and Berryshowed, phase singularities are very general topologicalproperties of wave fields and therefore intrinsically stableagainst small perturbations. It turns out that a descrip-tion in terms of phase singularities is extremely useful tounderstanding both the global and local properties of dif-fracted wave fields.

Using a laser beam, which generally has a Gaussian-beam profile, is of course a most common focusing ap-proach. Gaussian illumination of a lens results, in prin-ciple, again in a Gaussian beam behind the lens.6 Whatis important here is that such a Gaussian focal field con-tains no phase singularities, in contrast to the focal fieldproduced by uniform illumination. However, it is gener-ally known that singularities can disappear only when

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two singularities of opposite charge annihilate. There-fore an obvious question to ask is how the gradual tran-sition from uniform toward Gaussian illumination affectsthe presence and the spatial distribution of the phase sin-gularities in the focal field.

In this paper we address this question in detail, boththeoretically and experimentally. We study the distribu-tion of phase singularities in the focal region of a lens il-luminated by a Gaussian beam and truncated by the ap-erture of the lens. Theoretically, we discuss thestructure of the focal region, by means of numerical cal-culations based on diffraction theory, in terms of phasesingularities, and we study their distribution as a func-tion of the amount of truncation of the beam by the aper-ture. As we will show, in the case of partial truncation,the Airy rings reorganize themselves by means of acreation/annihilation process of the singularities. Thisprocess leads to extra dark rings outside the focal planeand, surprisingly, to structures on a subwavelength scale(in this paper we use the word subwavelength in thesense of arbitrarily smaller than 1 wavelength). As wewill see, this can be specially relevant in the case ofstrong focusing.

In our theoretical study of the focal field, we will relyheavily on the use of diffraction theory. We will not dis-cuss the different diffraction theories that we use in detailbut refer to the literature for more details. An overviewof the major theories can be found in Refs. 1 and 7. Forcompleteness, we mention Ref. 8, in which one of thepresent authors initiated the theoretical part of thisstudy, and Ref. 9, in which the preliminary results havebeen published.

Experimentally, we studied the three-dimensional (3D)

1998 Optical Society of America

Karman et al. Vol. 15, No. 4 /April 1998 /J. Opt. Soc. Am. A 885

intensity distribution of the focal region; to this end, weapplied a recently developed technique that enables us tomeasure this 3D intensity distribution with high resolu-tion and large dynamic range. The details of this tech-nique have been published in Ref. 10.

This paper is organized as follows: In section 2 we dis-cuss the nature of the problem and address the paraxiallimit by means of numerical calculations of the focal field,using scalar diffraction theory. In Section 3 we extendour analysis into the nonparaxial regime by means of vec-tor diffraction theory and discuss in detail the subwave-length aspects. In Section 4 we discuss the experimentsdone to verify the various predictions made for theparaxial regime. Finally, in Section 5 we summarize ourresults.

2. PARAXIAL CASEIn this section we state the nature of the problem, give atheoretical analysis in the paraxial regime, and presentthe numerical calculations by using scalar diffractiontheory. The situation that we address is depicted inFig. 1.

We consider the field produced by focusing a monochro-matic Gaussian beam of light (1/e amplitude width w, xpolarized), truncated by an aperture. The lens is as-sumed to be aberration free, meaning that the wave frontS after refraction is spherical and centered around thegeometrical focal point. The relevant parameters are thefocal length f, the lens radius a, and the half-apertureangle u. The numerical aperture NA of the configurationis defined as NA [ sin u, and the Fresnel number N asN [ a2/lf. In the paraxial regime (a ! f, i.e., NA ! 1),one component of the electromagnetic field is dominant,and a description in terms of a scalar wave field V(r, t) isadequate. In this regime V(r, t) satisfies the scalarwave equation

S ¹2 21

c2

]2

]t2DV~r, t ! 5 0 (1)

Fig. 1. Schematic focusing configuration. The lens is assumedto be aberration free; the focal distance f and the aperture radiusa are assumed to be large as compared with the wavelength.The origin of the coordinate system is placed in the geometricalfocal point. The incoming wave is assumed to be x polarized andpropagates in the positive z direction. Refraction at the lenscauses the E vector to rotate toward the focal point. ES is thefield on the wave front S after refraction. The aperture isplaced at z 5 2f. The wave vector of the incoming beam is de-noted by k.

and obeys specific boundary conditions. Assuming amonochromatic field V(r, t) [ u(r)exp(2iv t), the time-independent field amplitude u(r) has to obey the scalarHelmholtz equation

~¹2 1 k2!u~r! 5 0, (2)

with k [ v/c. In the remainder of this section, we willnumerically solve this equation in the paraxial limit.

A. Scalar Debye Diffraction TheoryWe will use the scalar Debye diffraction theory to find so-lutions of Eq. (2). The Debye theory has been shown tobe adequate in the limit kf @ p/sin2(u/2).11 In theparaxial regime (NA ' u ! 1), this is equivalent to thestatement that the Fresnel number N should be muchlarger than unity. In practical optical focusing configu-rations, this is often the case; the consequences of smallFresnel numbers are discussed in Appendix A. Webriefly remind the reader of the relevant formulas but re-fer to the literature for an extensive treatment of the De-bye theory.1,7

The Debye theory expands the focal field u(r) in a su-perposition of plane waves, which are the simplest exactsolutions of Eq. (2):

u~r! 5 exp~ik • r!, (3)

with the restriction kx2 1 ky

2 1 kz2 5 k2. Here k de-

notes the propagation vector of the plane wave. In prin-ciple, k is allowed to be complex; in that case Eq. (3) doesnot describe a traveling wave but represents an evanes-cent wave. The Debye approximation states that onlythose plane waves are taken into account that have a kvector lying in the cone formed by the aperture seen fromthe focal point:

u~x, y, z ! } EEV

U~kx , ky!

3 exp@i~kxx 1 kyy 1 kzz !#dkxdky , (4)

where V is the solid angle that the aperture subtends asseen from the focal point. Evanescent waves are obvi-ously not included in the angular superposition of relation(4). The relative weights U(kx , ky) in relation (4) can bedetermined from the field ua(x, y) inside the aperture,which we assume to be known:

U~kx , ky! }ua~2fkx /kz , 2fky /kz!

kz2 . (5)

When we assume that the input field has circular sym-metry around the optical axis [i.e., ua(x, y) 5 ua(r),where r [ Ax2 1 y2 is the distance to the optical axis],we can perform one integration in relation (4) analyticallyand the Debye integral reduces to

u~r, z ! } E0

k sin u ua~fkt /k !

kz2 exp~ikzz !J0~ktr!kt dkt ,

(6)

with

kz 5 Ak2 2 kt2. (7)


Fig. 2. Lommel field, which describes the focus of a lens with uniform illumination in the paraxial limit for the case NA 5 0.1 and f5 1000l. (a) Amplitude u(r) in the focal plane z 5 0 according to relation (9). (b) Intensity distribution in the (r, z) plane accordingto relation (8). The intensity ([uuu2) contours indicate intensities of 1021, 1022, ...; the intensity in the geometrical focus (0, 0) is nor-malized to 1.

From this expression one sees that the intensity distri-bution of the focal field is symmetric with respect to thefocal plane12; this property is typical for Debye theoriesand is lost in a Kirchhoff theory. Note that this integralsolves the scalar Helmholtz equation (2) for any value ofNA. The reason that we reject the solutions outside theparaxial limit is that in that case a scalar description isinadequate. In the paraxial limit, one can approximaterelation (6) and Eq. (7) through kz ' (k 2 kt

2/2k) andsin u ' u, leading to

u~r, z ! } E0

ku

ua~fkt /k !exp~ikzz !J0~ktr!kt dkt . (8)

In the remainder of this section, we will numericallyevaluate the paraxial Debye integral (8).

B. Various Input IlluminationsFirst, we discuss two focal fields that are well known fromliterature, the Lommel field and the Gaussian field, andanswer the question of how to relate them in terms ofphase singularities.

The Lommel field is the focal field of a uniformly illu-minated circular aperture ua(r) 5 1, calculated by meansof paraxial Debye theory. For this case the paraxial De-bye integral can be expressed analytically in Lommelfunctions.1 The focal field and the amplitude in the focalplane are plotted in Fig. 2 for the case NA 5 0.1.

As can be seen from relation (4), the field in the focalplane z 5 0 can be expressed analytically as the Fouriertransform in cylindrical coordinates of the circular aper-ture:

u~r, 0! }2J1~kur!

kur. (9)

This function corresponds to the well-known Airy ringpattern [see Fig. 2(a)]. The field has a bright spot in thecenter, surrounded by an infinite number of bright rings

and dark rings (of zero intensity) in between. The fieldoutside the focal plane is shown in Fig. 2(b) in the form ofan intensity contour plot, calculated by using relation (8).

The dark rings can be identified as phasesingularities,2,3 as illustrated in Fig. 3, where phase con-tour lines are plotted in the neighborhood of the first darkAiry ring. In the point of zero intensity, the phase con-tours join, indicating that the phase is undefined there,thus forming a phase singularity. This singularity is ac-companied by a phase saddle point S, where the gradientof the phase vanishes.

A phase singularity is a point around which the phaseincreases by 2pn over any closed path that encircles thesingularity; since the wave field is single valued, n is re-stricted to integer values.4 This number n, the topologi-cal charge, can be used to label a phase singularity. Thephase contours in Fig. 3 show that the dark Airy rings arestationary edge dislocations in the phase fronts, aroundwhich the phase surface is helical (phase vortex); thephase increases by 2p in one round trip, giving a topologi-cal charge of 11. The dislocation line coincides with thedark Airy ring, a closed circle in the focal plane, centeredaround the optical axis; the dark point in the figure is theintersection of this circle with a plane through the opticalaxis. It can be shown that, in general, singularities con-nected by the same phase contour line must have oppositecharge.13 In the case of the Airy rings, which all have thesame charge, the singularities are not connected, becauseof the presence of the phase saddle point S, which is dis-cussed in more detail in Ref. 4.

When considering the case of Gaussian illumination,ua(r) 5 exp(2r2/w2), with w ! a (no truncation), one cananalytically solve the diffraction integral in relation (4).The result is that the field after the lens is the Fouriertransform of a Gaussian, which is another Gaussianbeam. Obviously, phase singularities are absent in thiscase. An example of a Gaussian field distribution isshown in Fig. 4.

The field in each transverse plane has a Gaussian dis-tribution, as shown in Fig. 4(a). The width of the Gauss-


ian is a function of the z coordinate and has a minimumw0 in the focal plane z 5 0.6

We now ask ourselves the question of what happens inthe intermediate case: we illuminate the lens with aGaussian beam, having a 1/e amplitude width w, which isof the order of the lens radius a, i.e., the beam is partiallytruncated or apodized.14 We are not in the regime of uni-form illumination (w @ a), giving the Lommel field withphase singularities, and neither are we in the regime ofGaussian illumination (w ! a), giving a Gaussian-beamwaist without phase singularities. Will the focal fieldcontain phase singularities or not, and if so, what is theirspatial distribution? The answer to this question is to befound in the field outside the focal plane. Using theparaxial Debye diffraction integral in relation (8), we willcontinue the field in the (r, z) plane. As we have an in-terest in the Airy ring pattern or its remnants, we concen-trate on the region close to the focal plane but outside thegeometrical cone.

C. Reorganization of SingularitiesWe now describe the gradual conversion from a uniformamplitude toward a Gaussian by introducing a truncatedGaussian amplitude distribution in the aperture:

ua~r! 5 H exp~2r2/w2! for r < a

0 for r . a. (10)

In the limit a/w ↓ 0, the focal field is equivalent to theLommel field (Fig. 2). The intensity in the focal plane isthus the Airy ring pattern [relation (9)]. In the otherlimit, a/w → `, the intensity that is cut off by the aper-ture goes to zero, and we obtain a Gaussian distributionas in the waist of a paraxial Gaussian beam (Fig. 4). Thefield distribution in the focal plane in the intermediatecase (finite aperture, finite Gaussian width) is shown inFig. 5(a). For w @ a the Airy ring structure dominates.As the input amplitude starts to deviate from a uniformdistribution, the central spot grows. This is related tothe reduced spread of the beam in the aperture, whichgives a larger spread in the focal plane. This is accom-panied by a smaller distance between the Airy rings(marked A and B) close to the focus. The rings far from

the focal point do not move, but their intensity is reduced.Between a/w 5 1.563 and a/w 5 1.621, the innermosttwo zero points (A and B) approach and coalesce [see Fig.5(b)]. Beyond a/w 5 1.621 the innermost two dark rings(zero points) have disappeared. This process continueswith the next pair. As a/w becomes larger, the zeropoint that is then closest to the axis appears further fromthe axis r 5 0 and the next maximum has still lower in-tensity. In this way the Airy rings disappear, and thebeam distribution gradually approaches that of a Gauss-ian.

The question is now how to understand the disappear-ance of the rings in terms of phase singularities. Tostudy what happens with the Airy rings, we have calcu-lated the focal field in the (r, z) plane for the input fieldgiven by Eq. (10). In the limit a/w ↓ 0, we obtain theLommel field as depicted in Fig. 2(b). The opposite limit,a/w → `, gives the Gaussian beam as depicted in Fig.4(b). Two intermediate cases (finite aperture, finiteGaussian width) have been depicted in Fig. 6. Fromthese plots we can see that the zero points that disappear

Fig. 3. Enlargement of Fig. 2(b). Shown is the region close tothe first dark Airy ring at r ' 6.098l. Thick curves are con-tours of constant intensity, with adjacent lines differing by a fac-tor of 10 (and normalized to 1 in the focal point r 5 z 5 0).Thin curves are phase contours; the phase difference between ad-jacent phase contours is p/4. The point through which all phasecontours cross (the first Airy ring) is a phase singularity, and thepoint S slightly above it is a phase saddle point. The fact thateight phase contours spaced by p/4 collapse into the dark Airyring shows that the charge of the singularity is 11.

Fig. 4. (a) Amplitude of a paraxial Gaussian beam in the focal plane, with w0 /l 5 10; (b) (r, z) plane of this paraxial Gaussian beam.The intensity contours in (b) are, from the bottom, 1021, 1022, ..., relative to the focal point intensity.


from the focal plane can be found outside the focal plane.Analysis shows that this occurs by means of a creation/annihilation process, as illustrated in Fig. 7, which showsthe field near the first two Airy rings as a function of theparameter a/w.

If a/w is increased from zero to some finite value, thenthe two innermost singularities come closer together. Asthe two rings closest to the axis r 5 0 approach, the sec-ond one (labeled B) is split into three rings. This occursthrough the creation of two new singularity rings that

Fig. 5. (a) Field in the focal plane (z 5 0) in the case of a truncated Gaussian, with the use of Fraunhofer diffraction. (b) Close lookat the disappearance of two zero points in the Airy pattern; the first and second dark Airy rings are marked A and B, respectively.NA 5 0.1. The various curves in (a) and (b) correspond to different values of the ratio a/w.

Fig. 6. Two examples of truncated Gaussian beams at NA 5 0.1. For clarity, phase contours have been omitted, and only intensitycontours are shown, in the order 0.1, 0.01, ..., and normalized to 1 in the focal point. (a) a/w 5 1.515, enlarged in (b); (c) a/w5 1.818, enlarged in (d). The arrows in the enlargements point to phase singularities. The topological charges of the singularities

marked A, B, C, and D are, respectively, 11, 21, 11, and 11. This follows from considering the phase contours (not shown, but see Fig.13 below for a similar nonparaxial situation).


Fig. 7. Creation/annihilation process. Shown is the r –z plane in the focal region near the first two Airy rings (horizontally z/l, ver-tically r/l). NA 5 0.1. Intensity contours in the order 0.1, 0.01, ... are normalized to 1 in the focal point. From top left to bottom rightrowwise, the ratio a/w is increased from 1.370 to 1.667, showing the gradual transition from uniform toward Gaussian illumination.The labeling of the singularities is as in the other figures: A and B denote the Airy rings or its remnants, and C and D denote addi-tionally created rings. At a/w 5 1.471 creation of C and D occurs; at a/w 5 1.621 annihilation of A and B occurs.

have opposite topological charge. The two that haveequal topological charge lie outside the focal plane z 5 0and are labeled C and D. The third one lies in the focalplane and has the opposite topological charge (again la-beled B). For slightly larger values of a/w, this third sin-gularity annihilates with the innermost singularity in thefocal plane (labeled A).

During this process the various saddle points (as in Fig.3) accompanying the singularities behave similarly, asshown in more detail in Fig. 8. When the singularities Aand B approach each other, the saddle points, indicatedby crosses, first approach each other, then bounce off andend up outside the focal plane, and finally approach eachother again and annihilate together with the singulari-ties. This behavior of the saddle points during the anni-hilation of two singularities can be described analytically,as has been demonstrated in Ref. 15 (see, in particular,Fig. 4 of this reference). Various topological constraintsthat exist for singularities and saddle points have beendescribed in Ref. 16.

The result is thus that the innermost two Airy rings,which seem to have disappeared in Fig. 5, have in fact not

disappeared but have reorganized and end up outside thefocal plane. As long as these two extrafocal singularitiesare close to the focal plane, their presence is still visibleas a local minimum in the upper curve in Fig. 5(b). Ascan be seen from Fig. 9, this process continues with thenext pair of singularities in the focal plane. In the limitof large a/w (right-hand side of Fig. 9), many singulari-ties will have left the focal plane; they are found far fromthe optical axis at extremely low intensities.

Note that the intensity in the first bright ring is al-ready very small for moderate values of a/w. The fact

Fig. 8. Behavior of the saddle points during the annihilation oftwo singularities. The vertical line is the r axis, filled dots in-dicate the singularities A and B, and the crosses indicate thesaddle points. From left to right, A and B approach and anni-hilate in the point indicated by the open circle.


that these phenomena occur at very low intensity is pre-sumably the reason that they have never been recognizedbefore. For example, Fig. 12.38 of Ref. 7 shows the in-tensity distribution near the focus for a value of a/w suchthat the second singularity counted from the optical axisis about to split up into three, according to our calcula-tions. The lowest-intensity contour in that plot, how-ever, corresponds to an intensity that is well above the in-tensity of the first bright Airy ring.

In the paraxial examples above, we have calculated thefield in the case NA 5 0.1. Within the paraxial approxi-mation, the result for other values of the NA can be ob-tained with a trivial scaling: the transverse size scalesproportionally to (NA)21 and the longitudinal size scalesproportionally to (NA)22, as can be deduced from relation(8).

The reader may have gotten the impression that theparticular choice of a Gaussian illumination is essential;this is not the case. We have studied many amplitudedistributions ua(r) other than the truncated Gaussian.In most cases we could observe the reorganization pro-cess. The field patterns become, however, more compli-cated than in the Gaussian examples presented above, sothat interpretation becomes more difficult. Further-more, since Gaussian beams are widely used in practice,we have restricted our presentation to truncated Gauss-ian beams.

D. Subwavelength AspectsFrom the fact that, during the creation/annihilation pro-cess, the distance between singularities decreases to in-finitesimal values, one sees that this naturally leads tostructures on a scale arbitrarily smaller than 1 wave-length. This remarkable fact was already pointed out byBerry5,17: the topological nature of phase singularitiesensures their stability and their survival in the subwave-length regime. Of course, decreasing the distance be-tween neighboring singularities has its consequences forthe intensity in this region of the field: e.g., in Fig. 7 ata/w 5 1.621, the distance between A and B is smallerthan l/2, but the intensity in this region is also very small(,1029). As will become clear in Section 3, this intensity

Fig. 9. Location of the zero points in the focal plane as a func-tion of the truncation ratio a/w (with a kept constant), as calcu-lated with scalar paraxial Debye theory. NA 5 0.1. At thepoints at which two curves join, two phase singularities annihi-late. For example, at a/w 5 1.621, the singularities A and Bfrom Figs. 5(b), 6(b), and 7 annihilate. The left side of the figure(a/w ↓ 0) corresponds to uniform illumination, giving the Airypattern, and the right side (a/w → `) corresponds to a Gaussianbeam without singularities.

depends on the distance between the singularities andcan be increased by stronger focusing. Therefore we willpostpone a detailed discussion of the subwavelength as-pects to Subsection 3.E.

3. NONPARAXIAL CASEIt is clear that, outside the paraxial regime, vector aspectsof light are important on account of the large angles in theproblem. Therefore a scalar description is inadequate,and vectorial diffraction theory has to be used. The pur-pose of this section is to study the problem of Section 2outside the paraxial limit by means of vector calculations.Although a vector problem has in general a high degree ofcomplexity and the calculations are much more elaborate,we will show that the distribution of phase singularitiesremains a useful concept to describe the properties of thefield. To keep the discussion clear, this section is orga-nized along the same lines as those of Section 2, but, ofcourse, now with the use of vector theory.

A. Vector Debye Diffraction TheoryTo calculate the electric-field vector E in the focal region,we employ electromagnetic diffraction theory, based on avectorial equivalent of the Debye integral. A descriptionof this theory can be found in Refs. 7, 18, and 19. Again,this is allowed in the case of a large Fresnel number N,leading to a focal field that is symmetric with respect tothe focal plane. Since N 5 (NA)2f/l, one sees that forlarge NA the Fresnel number is of the order of f/l, whichis, of course, much larger than unity in most practicalcases. The case of small Fresnel numbers is discussed inAppendix A. When the polar coordinates q and w are de-fined in the usual way, the vectorial equivalent of the De-bye diffraction integral of relation (4) reads as

E~r! } E0

uE0

2p

ES~q, w!exp~ik • r!sin q dqdw,

(11)

where ES(q, w) is the electric field on the refracted wavefront S in Fig. 1, k(q, w) is the wave vector pointing fromthe wave front S toward the focal point, and the integra-tion extends over the wave front S in Fig. 1. Expressionsfor the field ES(q, w) can be found by applying theFresnel equations, describing the refraction of the electricfield at the surface of the lens. We consider the incomingE field to be x polarized, having a Gaussian-beam profileand propagating in the positive z direction, leading to thefollowing expression for ES(r):

ES~q, w! 5 E in~r!Acos q

3 S sin2 w 1 cos q cos2 w~cos w sin w!~cos q 2 1 !

sin q cos wD , (12)

E in~r! 5 exp~2r2/w2!

with r 5 f sin q and w @ l. (13)

In these equations E in(r) describes the Gaussian-beamprofile, the factor Acos q takes into account that the en-ergy flux of the incident plane wave front is projected ontothe spherical wave front S, and the terms inside the large


parentheses describe the rotation of the E vector at thelens surface as follows from the Fresnel equations. TheAcos q factor describing the energy projection correspondsto a so-called aplanatic focusing system. In the litera-ture two other types of projection are frequently encoun-tered: uniform and parabolic projection.7 We restrictourselves to the aplanatic energy projection; the othercases can be modeled by assuming a nonGaussian inputprofile on an aplanatic projection system. From Eq. (12)one clearly sees that in the paraxial limit (0 , q , uwith u ' NA ! 1), Ey and Ez are negligible when com-pared with Ex . To be precise, Eq. (12) then reduces to

ES~q, w! →NA!1

E in~r! 3 S O ~1 !

O ~u2!O ~u!

D , (14)

and one sees that only the x component of the electricfield survives; this is the justification of the use of scalartheory in Section 2. In fact, relations (11) and (14) areidentical to the scalar paraxial Debye diffraction integralwith the role of ua(r) played by E in(r). In general, ascan be seen from relation (14), the field components in thex, y, and z directions will be of the order of 1, (kw0)22,and (kw0)21, respectively, where w0 is the beam radiusin the focal region.20 This means that in the case ofstrong focusing (w0 ' l) the y and z components cannotbe neglected.

The w integration in the two-dimensional integral in re-lation (11) can be performed analytically, leading to ex-pressions involving Bessel functions:

E~r, u8, f8! } S I0 1 I2 cos~2f8!

I2 sin~2f8!

22iI1 cos f8D , (15)

with

I0 [ E0

u

E in~r!Acos q~sin q!~1 1 cos q!

3 J0~kr sin u8 sin q!

3 exp~ikr cos u8 cos q!dq, (16)

I1 [ E0

u

E in~r!Acos q sin2 qJ1~kr sin u8 sin q!


I2 [ E0

u

E in~r!Acos q~sin q!~1 2 cos q!

3 J2~kr sin u8 sin q!


where (r, u8, f8) denote the polar coordinates of the ob-servation point r. Similar expressions in terms of I0 , I1 ,and I2 can be found for the magnetic field, the Poyntingvector, and the energy density.7 From the dependence onthe azimuthal coordinate f8 in relation (15), one sees thatin general the focal field is not cylindrically symmetric,which complicates the presentation of the results. Forexample, the fields in the x –z plane differ from those inthe y –z plane. We will show the field distribution in thex –z plane only, since this plane turns out to be represen-tative for the structures found.

In the case of a vector field, it is not immediately clearwhat the proper generalization of a phase singularity is.Since we consider an x-polarized beam, Ex will be domi-nant and can be treated as a scalar field; furthermore,since in the paraxial limit Ex naturally goes over in thescalar wave field u in relation (6), phase singularities inEx are an obvious choice. At this point we mention theso-called disclinations, points at which the transversepart of E completely vanishes and the direction of the po-larization ellipse is singular. Disclinations have beenput forward by Nye as a vector generalization of a scalarphase dislocation.21 From relation (15) we have Ey 5 0in the x –z plane, which means that zeros of Ex in the x –zplane coincide with the disclinations introduced by Nye.

B. Various Input IlluminationsWe assume a lens with NA 5 0.9 (u 5 64°), f 5 1000l,and a 5 2065l. Uniform illumination, E in(r) 5 1, leadsto a focal field as depicted in Fig. 10.

Fig. 10. Field distribution in the x –z plane in the case of uniform illumination: NA 5 0.9, f 5 1000l, and a 5 2065l. (a) Contourlines of total energy density (E • E* 1 B • B* ), in the order 0.5, 0.2, 0.1, etc., normalized to 1 in the focal point. (b) Ex : thick curvesare curves of constant intensity uExu2, with adjacent curves differing by a factor of 10 and normalized to 1 in the focal point; thin curvesare curves of constant phase, with adjacent curves differing by p/4.


Shown are the total energy density (E • E* 1 B • B* )and the intensity and phase contours of the x componentof the E field. The difference between the two is a resultof the fact that y and z components cannot be neglected.One clearly sees the familiar Airy pattern in the focalplane, as formed by the zeros of Ex , just as in the scalarparaxial case. Note that E • E* 1 B • B* has no zeros.In the remainder of this section, we will concentrate onthe x component as shown in Fig. 10(b).

The case of a Gaussian illumination, E in(r)5 exp(2r2/w2) with w ! a, is shown in Fig. 11.

We conclude that, outside the paraxial limit, Gaussianillumination (with w ! a, i.e., no truncation) leads to afocal field that contains no singularities (zeros of Ex).From this we expect that in the intermediate case (w' a) a reorganization process will occur, similar to thatin the paraxial case.

C. ReorganizationNow we consider an intermediate case: we use the samelens, but the Gaussian beam has now a 1/e amplitude ra-dius of w 5 570l and is thus truncated by the lens (a/w5 3.62). The amplitude and the phase of Ex in the focalregion are displayed in Figs. 12 and 13.

As illustrated in Figs. 12 and 13, the Airy rings reorga-nize themselves in the same way as in the paraxial case(Fig. 6): starting from a uniform illumination (a/w ↓ 0),we have the Airy rings A and B in the focal plane. Whena/w increases, B is split into three (B, C, and D); uponfurther increasing a/w, we see that A and B approacheach other, and C and D move away from the focal plane.This stage is depicted in Fig. 13. Finally, at still largera/w, A and B annihilate. The vorticity of the phase con-tour curves encircling the singularities defines the associ-ated topological charge (the charges of A, B, C, and D, arerespectively 11, 21, 11, and 11). Note that thecreation/annihilation process conserves the total topologi-cal charge.

This reorganization process occurs on a subwavelengthscale: in Fig. 13 the singularities are separated by dis-

Fig. 11. Distribution of Ex in the x –z plane in the case ofGaussian illumination: NA 5 0.9, f 5 1000l, a 5 2065l, w5 200l, and a/w 5 10.3. Shown are curves of constant inten-sity ([uExu2), with adjacent curves differing by a factor of 10 andnormalized to 1 in the focal point.

tances of approximately 0.15l. Note that the intensityuExu2 in this region of the focal field is very low: the in-tensity midway between the singularities A and B is ap-proximately 2 3 1026. Although low, it is much higherthan that in the paraxial case shown in Fig. 6: increas-ing the NA also increases the intensity of the region inwhich the subwavelength structures are embedded. Thisaspect will be discussed in detail in Subsection 3.E.

D. Transition from the Paraxial to the Nonparaxial CaseThe reorganization process seems to be independent ofthe NA of the system. To study this in more detail, wehave plotted in Fig. 14 the positions of the singularitiesalong the x axis as a function of the ratio a/w for differentvalues of NA.

Of course, for NA50.1, we recover the paraxial result ofFig. 9, as can also be seen by making the paraxial ap-proximation in relation (15), leading to

Fig. 12. Distribution of Ex in the x –z plane in the case of a par-tially truncated Gaussian beam: NA 5 0.9, f 5 1000l, a5 2065l, w 5 570l, and a/w 5 3.62. Thick curves are curvesof constant intensity ([uExu2), with adjacent curves differing bya factor of 10 and normalized to 1 in the focal point. Thin curvesare curves of constant phase, with adjacent curves differing byp/4. A series of dark (Airy) rings appears in the focal plane; theregion around the first dark ring is enlarged in Fig. 13 below,showing the reorganization process.

Fig. 13. Enlargement of a part of Fig. 12. The four phase sin-gularities are labeled A, B, C, and D, as in Fig. 6(b), and areseparated by distances of '0.15l. Singularities A and B areremnants of the Airy rings, and singularities C and D are newlycreated singularities and have moved away from the focal plane.The vorticity of the phase contour lines around the singularitiesindicates the topological charge; the charges of A, B, C, and Dare, respectively, 11, 21, 11, and 11.


Ex~x; y 5 z 5 0 ! →NA!1 E

0

u

E in~r!J0~kxq!q d q, (19)

which is identical to the paraxial relation (8) for z 5 0.One sees that in the gradual transition from uniform

toward Gaussian illumination the Airy rings reorganizethemselves by means of a creation/annihilation process,independent of the NA: the only difference that the valueof NA makes is that it alters the a/w value at which thereorganization process starts. Paraxially, the reorgani-zation starts at a/w ' 1.5, whereas for NA50.9 the pro-cess starts at a/w ' 3.4; in both cases the larger a/w is,the more the rings reorganize themselves until all ringshave left the focal plane. The fact that the reorganiza-tion starts at larger a/w for larger NA can be understoodfrom the presence of the factor Acos q in Eq. (12) describ-ing the aplanatic energy projection. This factor multi-plies the Gaussian-beam profile E in(r) and thereby effec-tively decreases its width (the larger the NA, the strongerthis effect), thus leading to a larger a/w value.

The fact that the reorganization process is independentof the NA does not mean that the vector character of lightis irrelevant. So far, we have concentrated on the x com-ponent only; however, since the y and z components of thefield are nonzero, the total intensity does not show thesame behavior as that of the x component. To illustratethis, we have plotted in Fig. 15 the different field compo-nents along the x axis for the specific case of Figs. 12 and13 (where reorganization is in progress).

From Fig. 15 one sees that the total energy density ex-hibits no subwavelength structure (neither does the elec-tric energy density, not shown).

E. Subwavelength AspectsSubwavelength structures in a wave field are well knownin the form of evanescent waves (in which the wave vectoris complex). However, these structures are limited to thenear field of a material object; this is exploited in tech-niques such as near-field optical microscopy, where onescans a probe through the near field of an object to makean image on a subwavelength scale. We have found thata wave field can contain structure in its far field on a scalesmaller than 1 wavelength; this seems to be fundamen-tally different from subwavelength structure that is dueto evanescent waves.

On the one hand, the fact that subwavelength struc-tures can exist in the far field can be understood from thetopological nature of the singularities: the creation/annihilation process allows for an arbitrarily small dis-tance between the edge dislocations A 2 B and B 2 C2 D in Figs. 6(b), 7, and 13. On the other hand, thisseems to be in conflict with Fourier theory: wave fields ofwavelength l cannot show structures oscillating on ascale smaller than l, since this is the highest Fouriercomponent present. As pointed out by Berry,17,22 topo-logical structures separated by distances much smallerthan the wavelength of the light can be described by so-called superoscillating functions, i.e., functions that oscil-

Fig. 14. Positions of the singularities (zeros of Ex) along the x axis as a function of the truncation ratio a/w, for different values of NA,as calculated with vector Debye theory. The top left plot corresponds to the paraxial limit (NA 5 0.1, as in Fig. 9); the other plots showthe results for NA 5 0.3, 0.7, and 0.9. As in Fig. 9, the joining of two curves indicates annihilation of two singularities.


late arbitrarily faster than their highest Fourier compo-nent. The price paid for having such superoscillations isthat the field amplitude becomes very low in this region ofthe field. See Appendix B for a brief description of sup-eroscillating functions.

We will show in this subsection that the resolution ofthese subwavelength structures, by which we mean thedistance between neighboring singularities, has conse-quences for the intensity of the region in which the singu-larities are embedded. We will focus on the question ofwhat determines the value of the embedding intensityand address in particular the role of the NA of the focus-ing system.

As can be deduced from Fig. 13, the intensity betweentwo phase singularities has a saddle point; the saddle-point intensity is a good measure for the visibility of thesubwavelength structures. To investigate the conse-quences of reducing the distance between the phase sin-gularities for the saddle-point intensity, we have calcu-lated the intensity in the saddle point as a function of thedistance.

Figure 16 shows the intensity of the x component of theelectric field in the saddle-point between the singularitiesA and B as a function of their separation Dx. This dis-tance was changed by changing the width w of the inputGaussian beam (in the range w 5 560l to 580l) whilekeeping NA50.9 and f 5 1000l constant. The curveobeys a simple power law: I } (Dx)4. This result can beeasily understood as follows: the field amplitude near anisolated singularity with charge 61 is locally linear in thedistance to the singularity.4 Between the two singulari-ties, the field has to have a maximum. The simplest fieldfulfilling these requirements is described by a parabola,having an amplitude quadratic in the distance. Thisquadratic term is of course the first higher-order term in aTaylor expansion; in fact, we have checked this by com-putation for distances as displayed in Fig. 16 and foundthat the field between the singularities A and B is almosta perfect parabola. The maximum of the parabola coin-cides with the saddle point. Since the intensity is pro-portional to the square of the field, this gives I } (Dx)4.This result can be extended to phase singularities ofhigher topological charges, revealing the general rule I} (Dx)2(umu1unu), where m and n are the topological

Fig. 15. Different field components along the x axis for the casedepicted in Figs. 12 and 13: the x component of the electricfield, uExu2 (dashed curve); the z component uEzu2 (solid curve);and the total energy density E • E* 1 B • B* (dotted curve).E • E* has been normalized to 1 in the focal point. The first twozeros of Ex are separated by a distance of 0.15l and correspond tothe singularities A and B in Fig. 13.

charges of the singularities; that is, the higher are thecharges involved, the faster the field decays.

We observed in Fig. 14 that the reorganization processis quite independent of NA. What does change is the in-tensity of the region in which the subwavelength struc-tures are embedded. Even in the paraxial limit (NA!1),subwavelength structure remains present, although atextremely low intensities; the only result of lowering NAis a decrease of the intensity in the saddle point betweenthe singularities. To investigate this effect, we variedNA while keeping Dx 5 0.15l constant; the result isshown in Fig. 17.

We observe again a power law: I } (NA)4. It illus-trates the major difference between the paraxial and non-paraxial cases: the NA determines the intensity of theregion of the field in which the subwavelength structuresare embedded. From an experimental point of view, it isclear that to observe these structures in the subwave-length regime, it is necessary to strongly focus the beamin order to maximize the intensity in the saddle point.

By combining the two results, we find I } (NA 3 Dx)4

for the saddle-point intensity. The fact that the exponentof the two power laws is the same suggests a common ori-gin of the two power laws. However, the fourth power inI } (Dx)4 has its origin in the local topological properties

Fig. 16. Calculated intensity I [ uExu2 in the saddle point be-tween the neighboring phase singularities A and B as a functionof their distance Dx. The distance was changed by slightlychanging the width w of the beam (while keeping NA 5 0.9 con-stant). Each intensity value was divided by the intensity in thefocal point, so the values plotted are relative intensities. Notethe double logarithmic scale. The line is a linear fit to the cal-culated points and has a slope of 4.02 6 0.03.

Fig. 17. Calculated intensity I [ uExu2 in the saddle point be-tween two neighboring phase singularities A and B as a functionof NA (with Dx 5 0.15l kept constant). Each intensity valuewas divided by the intensity in the focal point, so the values plot-ted are relative intensities. Note the double logarithmic scale.The line is a linear fit to the calculated points and has a slope of4.1 6 0.1.


of the phase singularities, whereas the fourth power in I} (NA)4 is related to the global structure of the field, asdictated by the NA.

4. EXPERIMENTAL VERIFICATIONIn this section we describe an experimental study of thereorganization effects. In Sections 2 and 3, we showedtheoretically that the most pronounced effect of the reor-ganization process is the existence of extra dark rings out-side the focal plane; they occur in both the paraxial andnonparaxial regimes. These extrafocal dark rings seemthe most promising aspect for experimental verification.To this end, we developed an experimental technique tomeasure the 3D intensity distribution of a focal field; ourgoal is to measure an intensity map as in Fig. 6. We firstdescribe the experimental setup and technique used toperform measurements of the 3D intensity distribution ofthe focal field. Then we discuss the experimental resultsand compare them with theory.

A. Experimental SetupThe details on the experimental technique to measure 3Dintensity distributions in the neighborhood of a paraxialfocus have been published elsewhere10; therefore we willgive only a brief description.

As shown in Fig. 18, we illuminate an apertured lenswith a linearly polarized Gaussian laser beam and place atwo-dimensional CCD image sensor in the focal region tomake an image of the beam profile at a certain z coordi-nate. This is repeated many times for different z coordi-nates (typically at 500 z coordinates); to this end, theCCD sensor was mounted on top of a translation stage, al-lowing us to move the sensor along the z axis. From thedata obtained in this way, we construct an intensity mapof the r–z plane. This map can then be compared withthe various theoretical results as shown in the previoussections.

For successful implementation of this concept, specialcaution should be taken to ensure the mechanical stabil-ity of the setup. When the beam-pointing stability of thelaser and the mechanical stability of the translation stageare good enough, intensity maps accurate to the size of 1pixel on the CCD chip (9 mm 3 9mm) can be obtained.Using a CCD image sensor having a large intrinsic dy-namic range, combined with overexposing to extend thisdynamic range, we were able to produce intensity mapsshowing a dynamic range of more than 5 orders ofmagnitude.10 To increase the signal-to-noise ratio of thesignal, we averaged the intensity distribution in the azi-muthal direction; this is allowed because the intensitydistribution is circularly symmetric in the paraxial limit.A further increase in signal-to-noise ratio can be accom-plished by averaging multiple pictures at each z position.To achieve high-resolution intensity maps, the use of thistechnique is again restricted to paraxial circumstances:the size of the diffraction pattern scales proportionally to(NA)21 in the transverse direction and proportionally to(NA)22 in the longitudinal direction. Therefore a smallvalue of NA is desired. Pixel sizes of the order of 9 mmlimit the NA to values below 1022. However, a value be-low 1023 leads to a longitudinal extent of the diffraction

pattern that exceeds 1 m, which is unpractical; thereforethe NA is limited to values of NA 5 1023 to 1022. Thismakes the technique especially suitable for verification ofthe reorganization phenomena in the paraxial limit, asdiscussed in Section 2.

From the fact that the pixel size of the CCD chip (9 mm)is much larger than the wavelength of the laser beam (l5 632.9 nm), it is clear that the smallest detail that ourmethod can resolve is 1 pixel. This is sufficient to detectthe presence of the extra singularities outside the focalplane. The subwavelength aspects of the reorganizationprocess cannot be verified; this would require a detectormuch smaller than 1 wavelength, e.g., a single fluorescentmolecule.23

B. Experimental ResultsFirst, we measured the positions of the singularities inthe geometrical focal plane as a function of the truncationratio a/w. As an aperture we used an iris diaphragm, al-lowing an easy adjustment of its size; in this way we canadjust the a/w ratio. The fact that the diaphragm wasnot perfectly circular (instead, it is a dodecahedron,whose ‘‘radius’’ varies by 2%) was not problematic becausewe averaged the intensity distribution in the azimuthaldirection. The result is shown in Fig. 19.

The position of the singularities is shown in Fig. 19 as afunction of a/w; this ratio was varied by changing a. A

Fig. 18. Experimental setup. The following acronyms areused: P 5 polarizer, SF 5 spatial filter/telescope, L 5 lens, A5 aperture, and CCD 5 CCD image sensor. The origin of thecoordinate system is located at the geometrical focal point.

Fig. 19. Experimental result: positions of the singularities inthe focal plane as a function of the ratio a/w (with w kept con-stant). The conditions are l 5 632.9 nm, f 5 1 6 0.02 m, w5 1.74 6 0.04 mm, a 5 0.8 to 3 mm, NA ' 2 3 1023, and N' 6. The circles are the experimental results, and the curvesare the corresponding theoretical results. The vertical axisshows ar, where a is the aperture radius and r is the distance tothe optical axis. The first two Airy rings are labeled A and B.


consequence is that the NA also changes; the highera is, the higher NA is and the closer the singularitiesare to the optical axis. So the trend r } 1/NA } 1/a isunderstandable; to correct for this trend, we have shownvertically in Fig. 19 ar, which is more or less constant.We see that for increasing a/w the first two Airy rings(A and B) approach, until at a/w ' 1.6 they annihilate.From Fig. 19 one sees the good agreement with theory(cf. Fig. 9). From this result it is clear that thetwo singularities A and B disappear from the focalplane.

The next step is to check whether these singularitiescan indeed be found outside the focal plane. To this end,we measured the 3D intensity distribution in the neigh-borhood of the focal point.

In Fig. 20 the experimentally obtained intensity distri-bution in the focal region is shown in the form of an in-tensity contour plot. The figure contains 500 3 100 pix-els in, respectively, the horizontal and vertical directions.Linear interpolation was used to draw contour lines be-tween pixels; in this way the underlying grid is hardlyvisible. The ratio a/w 5 0.59 is such that the familiarAiry pattern in the focal plane can be observed: sevensingularities can be seen in the plane z 5 0. The asym-metry with respect to the focal plane is due to the finiteFresnel number N 5 8.5.24 Remarkable is the almost to-tal absence of noise in the various contours; this is causedby the averaging in the azimuthal direction and the use ofmany (500) pictures. The use of overexposed imagesleads to an intensity map showing a dynamic range of 5decades.

To observe the reorganization phenomenon, we concen-trate on the boxed region in Fig. 20 near the first two Airyrings. For different values of a/w, this region is shownin Fig. 21.

From Fig. 21 we conclude that in the gradual transitionfrom uniform toward Gaussian illumination of the lens L,the Airy rings, labeled A and B, do indeed disappear fromthe focal plane z 5 0. By means of a creation/annihilation process, extra singularities (labeled C and D)are created outside the focal plane, and the remnants of

the Airy rings annihilate. These experimental resultsare in full agreement with the theory as discussed in Sec-tion 2.

The fact that the experimentally obtained intensity dis-tributions in Figs. 20 and 21 are somewhat asymmetricwith respect to the focal plane can be understood from thefact that, on account of the finite Fresnel number (N' 8), the Debye theory is not strictly applicable. In-stead, a Kirchhoff diffraction theory should be used (seeAppendix A). We have found that the experimental re-sults are in good agreement with computations based onthe Kirchhoff theory (not shown, but see Ref. 10).

It should be noted that the exact size of the aperture isquite important. From Fig. 7 one sees that to proceedfrom the Airy pattern toward the situation in which thesingularities in the focal plane have annihilated (leavingonly singularities outside the focal plane), it is sufficient

Fig. 20. Experimental result: the intensity distribution in theneighborhood of the focal point. Shown are contours of constantintensity, normalized to 1 in the geometrical focal point (r 5 z5 0). The conditions are f 5 0.6 6 0.01 m, a 5 1.86 0.02 mm, w 5 3.0 6 0.1 mm, NA 5 3 3 1023, N 5 8.5, anda/w 5 0.59. Clearly, one sees the Airy rings in the focal plane(z 5 0). The reader should compare this figure with the theo-retical result of Fig. 2. We concentrate on the boxed region nearthe first two Airy rings.

Fig. 21. Experimental result: the boxed region of Fig. 20 for two values of a/w. The conditions are f 5 1 6 0.02 m and w 5 1.466 0.05 mm. (a) a/w 5 1.44 (a 5 2.1 6 0.05 mm). Beside the Airy rings A and B, the extra singularities C and D are clearly visibleoutside the focal plane. (b) a/w 5 1.6 (a 5 2.3 6 0.05 mm). Only the singularities C and D are present. A and B have annihilated.The reader should compare the plots in (a) and (b) with the theory in Fig. 7.


to increase the ratio a/w by only 15%. Therefore, to ob-serve the intermediate stages of the reorganization pro-cess, one should adjust the aperture size carefully; wefound that this was still possible by using a dodecahedraldiaphragm (circular to 2%).

From our experience we found that special attentionought to be paid to two experimental problems: align-ment and the cover glass of the CCD sensor. First, it iscrucial that the alignment of the setup is such that thebeam profile is circularly symmetric everywhere; astig-matism caused by misalignment of the optical elementsshould especially be avoided. Only then can one averagein the azimuthal direction, which is necessary to obtain alarge enough signal-to-noise ratio to observe the detailedsingularity structure as in Fig. 21. We found that resultsobtained by averaging over a small sector (instead of overthe full 360°) were of poorer quality, which, however, canbe compensated for by making more pictures at each z po-sition and subsequent averaging. We used this last pro-cedure in Fig. 21, since we found it to be a practical com-promise. A second problem is that most CCD chips areequipped with a cover glass to protect the chip from theenvironment. This cover glass causes undesired reflec-tions: when the laser beam hits the CCD chip, the back-reflected beam contains a diffraction pattern of the chipitself (which has square symmetry because of the squarepixels). This diffraction pattern is reflected by the coverglass back onto the CCD chip. Thus the result is that theCCD chip registers a superposition of this undesired dif-fraction pattern and the laser beam itself, instead of onlythe beam profile. Normally, when one is not interestedin the very-low-intensity regions of the field, this effectposes hardly any problems. But we have a special inter-est in the regions of low intensity; furthermore, since westrongly illuminate the CCD chip (to overexpose it in or-der to increase the dynamic range), we found this effect tobe troublesome. Three solutions to this problem exist:(1) removing the cover glass and leaving the CCD chip un-protected, (2) antireflex coating the cover glass, and (3)adjusting the NA of the lens L (within the range1023 –1022) such that the diffraction pattern that is dueto the chip itself does not coincide with the region nearthe first two Airy rings. Furthermore, the main maximaof the undesired diffraction pattern are located on axesparallel and perpendicular to the square grid structure ofthe pixels on the chip; avoiding these regions in the azi-muthal averaging procedure also helps to eliminate thisproblem. This last option was chosen to obtain the re-sults in Figs. 20 and 21.

At first glance one could expect that problems such asthose mentioned above could have a devastating effect onthe fine singularity structure. We believe, however, thatthe topological nature of the singularities ensures theirstability under all kinds of changes in the boundary con-ditions. Therefore we expect (and qualitatively observed)the singularity structure to be highly stable against lensaberrations, poor beam quality, noncircular symmetry ofthe aperture, misalignment, etc.

5. CONCLUSIONSWe studied the effects of truncation of a Gaussian beamon the structure of the focal field. We showed that the

concept of phase singularities is useful in analyzing the3D structure of the field. The dark rings in the well-known Airy pattern are examples of phase singularities.We found that, in general, when proceeding from uniformtoward Gaussian illumination of the lens, the well-knownAiry rings reorganize themselves by means of a creation/annihilation process of phase singularities, independentof the NA. The most pronounced effect is the occurrenceof extra dark rings outside the focal plane of the lens.We developed an experimental technique to map the 3Dintensity distribution of the focal field to verify these pre-dictions in the paraxial regime. The experiments con-firm the existence of extra dark rings outside the focalplane, and their reorganization was observed.

The consequences of this reorganization process aremost interesting outside the paraxial limit, since theynaturally lead to subwavelength structures in the farfield. We clarified the connection with similar results ob-tained by Berry; our results constitute the translation ofBerry’s work into optics.17,22 Although, in principle,these subwavelength structures can be present in theparaxial limit as well, a large NA is necessary to bringthem into a region of larger (measurable) intensities.Typically, structures on a scale of Dx 5 0.15l exist atrelative intensities of uExu2 ' 2 3 1026 for a NA of 0.9.The relation between the various parameters is uExu2

} (NA 3 Dx)4.

APPENDIX A: CONSEQUENCES OF SMALLFRESNEL NUMBERSThis appendix discusses the effect of small Fresnel num-bers on the results obtained in Sections 2 and 3. One canquestion the validity of these results in the case of a smallFresnel number, since then the use of the Debye theory isinappropriate; instead, a Kirchhoff theory should be used.To this end, we computed all cases of Sections 2 and 3again by using Kirchhoff theory instead of Debye theory.We include this generalization as an appendix, since theconsequences of a small Fresnel number are relativelyminor and do not affect the essence of our results.

1. Scalar CaseThe most important difference between the Kirchhofftheory and the Debye theory is that the symmetry of thefocal field with respect to the focal plane is lost. This isaccompanied by the so-called focal shift24: the point ofmaximum intensity lies not in the geometrical focal pointbut between the focal plane and the lens. A descriptionof the scalar Kirchhoff theory can be found in Refs. 7 and25. Maps showing the field distribution for differentFresnel numbers can be found in Refs. 7 and 24. As weare interested in the spatial distribution of the phase sin-gularities, we show in Fig. 22, for a Fresnel number N5 3, this distribution in the case where the first two Airyrings have reorganized into four singularities.

One sees that the symmetry with respect to the focalplane z 5 0 is severely broken: the figure is transverselycompressed toward the side of the lens. This deforma-tion is stronger the lower the Fresnel number is. Impor-tant here is that the (extra) singularities are still present:the topology of the pattern remains unchanged. Further-


more, in the limit of small distances between the singu-larities B 2 C 2 D, the asymmetry of the pattern ishardly noticeable. From this we conclude that the de-scription with the Debye theory in Section 2 is accuratewith respect to the reorganization process.

2. Vector CaseOutside the paraxial limit, the case of low Fresnel num-bers is somewhat artificial: from N 5 (NA)2f/l, onesees that for large NA one has N ' f/l, which is, in prac-tice, much larger than unity. However, for completeness,we repeated the vector calculations with the use of aKirchhoff vector theory. For details on this theory, werefer to the work of Visser and Wiersma18,19; their theorycan be used to study lens aberrations as well. These cal-culations lead us to the same conclusion as that in thescalar case in Appendix A.1: the focal field loses its sym-metry with respect to the focal plane, but the topologicalfeatures with respect to the spatial distribution of thephase singularities remain unchanged.

APPENDIX B: SUPEROSCILLATINGFUNCTIONSTo make clear how it is possible that functions can oscil-late faster than their highest Fourier components, con-sider the following one-dimensional example:

f~x, A, d! 5 E2`

`

exp@ik~u !x#expF21

d2 ~u 2 iA !2Gdu,

(B1)

k~u ! 5 cos ~u !, (B2)

where d is small and A is real and positive. It is a super-position of plane waves, with the second exponential act-ing as a weight factor. Note that uk(u)u < 1 for real u;i.e., the highest Fourier component is k 5 1. The idea isthat for small d the second exponential acts like a deltafunction that selects u 5 iA. Then f(x) oscillates asf(x) ' exp(iKx), which is much faster than exp(ix), sinceK 5 k(iA) 5 cosh (A) can be much larger than unity. Inthis way one can construct functions that oscillate arbi-trarily faster than their highest Fourier component in an

Fig. 22. Intensity distribution in the case of the low Fresnelnumber N 5 3, where NA 5 0.1 and a/w 5 1.60. This is to becompared with the Debye result (N 5 `) of Fig. 7. The inten-sity is normalized to 1 in the geometrical focal point, and the con-tour lines are shown in the order 1, 0.1, 0.01, ... .

arbitrarily long interval. For more details on superoscil-lating functions, we refer to the work of Berry.17,22

Note added in proof. After completion of this paper, apaper appeared by Totzeck and Tiziani,26 discussingphase singularities in the near field of a structured sub-strate. As in the case of our far-field singularities, theauthors predict creation/annihilation processes.

ACKNOWLEDGMENTSThis work is part of the research program of the Founda-tion for Fundamental Research on Matter (FOM) and wasmade possible by financial support from the NetherlandsOrganization for Scientific Research (NWO). We also ac-knowledge support from the European Union under ES-PRIT contract 20029 (ACQUIRE) and TMR contractERB4061PL95-1021 (Microlasers and Cavity QED).

Address all correspondence to G. P. Karman; e-mail:[email protected].

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Airy pattern reorganization and subwavelength structure in ...web.physics.ucsb.edu/~quopt/airy.pdf · Airy pattern reorganization and subwavelength structure in a focus G. P. Karman,