modeling illumination-mode near-field optical microscopy of au nanoparticles

704 J. Opt. Soc. Am. A/Vol. 18, No. 3 /March 2001 Liu et al.

Modeling illumination-mode near-field opticalmicroscopy of Au nanoparticles

Ansheng Liu,* Adel Rahmani, Garnett W. Bryant,† Lee J. Richter, and Stephan J. Stranick

National Institute of Standards and Technology, Gaithersburg, Maryland 20899

Received February 11, 2000; revised manuscript received September 29, 2000; accepted October 2, 2000

We present a theoretical analysis of near-field scanning optical microscopy (NSOM) images of small Au par-ticles made in the illumination mode. We model the metal-coated fiber tip as a thin disk consisting of a glasscore and an aluminum coating. An external field locally illuminates the tip core. We solve for the local fields,including interactions between the tip and the Au particles, by use of the coupled dipole method and calculatethe optical signal collected in the far field. We also determine the tip field, in the absence of the particle, forvarious tip sizes with different metal-coating thicknesses. Calculated tip fields and simulated images arecompared with those obtained with the Bethe–Bouwkamp model, a commonly used simple model for the tipfield. Calculated line scans of the NSOM images of Au particles depend strongly on the tip aperture size andmetal-coating thickness. For blunt tips with a thick metal coating and sharp tips with a much thinner coat-ing, our thin-disk model reproduces the key features of measured NSOM images. Line scans calculated withthe Bethe–Bouwkamp model cannot describe the tip dependence of the experimental images. Tip fields ob-tained from the thin-disk model show significant enhancement beneath the metal coating and a broader fielddistribution perpendicular to the polarization. Tip fields obtained with the Bethe–Bouwkamp model do notshow these effects. Differences in the line scans for these two models are correlated to the differences betweenthe tip fields for the two models. These differences occur because only the disk model accounts for a finitemetal coating. © 2001 Optical Society of America

OCIS codes: 180.0180, 290.5850, 180.5810, 260.2110.

1. INTRODUCTIONNear-field scanning optical microscopy1 (NSOM) is a new,emerging optical probe technique that has been developedto probe selectively, for example, single molecules,2,3

semiconductor nanostructures such as quantum wells,4

wires,5,6 and dots,7 and photonic materials (nanochannelglass arrays).8 Spatial resolution in conventional far-field optical spectroscopies, such as transmission, absorp-tion, and photoluminescence spectroscopies, is diffractionlimited to .l/2 (l is the wavelength of the light invacuum). NSOM can provide spatial resolution muchbetter than l/2 because the highly spatially localized nearfield of an NSOM tip is used as the probe. Inillumination-mode NSOM experiments, the nanometer-scale sources are typically metal-coated fiber tips with anaperture that is much smaller than l. The sample isplaced in the near field of the tip to exploit the strong lo-calization of the tip field. The optical signal is recordedin the far-field region. The NSOM images are deter-mined by the optical response of the sample to the nearfield of the tip and by the tip–sample mutual interactions.To interpret NSOM images correctly, the near field of thetip must be modeled carefully and the tip–sample inter-actions must be understood.9 Various models have beenused to represent tip fields. The Bethe–Bouwkamp (BB)solutions for the field near a small aperture in an infinite,perfectly conducting, thin, flat screen10,11 have been usedextensively as a simple model for the near field of a metal-coated tip. For example, one of the first uses of the BBmodel was by Betzig and Chichester2 to interpret NSOMimages of single molecules. We have employed the BBmodel to analyze transmission NSOM images of

0740-3232/2001/030704-13$15.00 ©

nanochannel glass arrays.12 In the BB model the tipfield is determined solely by the aperture size of the tip.Any dependence of the NSOM images on other tip param-eters, such as coating thickness or tip shape, cannot be ac-counted for by the BB model. Tip fields can also be mod-eled by the discrete dipole approach.13–17 In thisapproach, tip fields are determined by representing thetip as a set of coupled, discrete, radiating dipoles. Thetip field is the self-consistent response of these dipoles toan external driving field. In contrast to the BB model,the discrete dipole approach can account for the depen-dence of the tip field on tip size and shape if the distribu-tion of discrete dipoles accurately models the tip.

In this paper we compare theory and a well-characterized NSOM imaging experiment to test the va-lidity of the discrete dipole model for tip fields and to de-termine the influence of tip size and shape on NSOMimages. We analyze a simple model for the tip. We con-sider a set of coupled, discrete dipoles distributed in athin disk to model the response of the end of a metal-coated NSOM tip. Dipoles with different polarizabilitiesare used to model the glass core and metal coating of thetip. The effects of the finite thickness of the metal coat-ing are included by considering disks with a finite radius.We simulate a simple imaging experiment in which Auparticles on a glass surface are probed in illumination-mode NSOM and the transmitted light is collected to pro-duce images. To model this experiment, we also repre-sent the Au particle as a set of coupled dipoles. The self-consistent response of the tip to a driving field and theinteraction between the tip and particle are determinedin the coupled dipole approach.

2001 Optical Society of America

Liu et al. Vol. 18, No. 3 /March 2001/J. Opt. Soc. Am. A 705

By comparing detailed calculations with experimentsdone with tips of different sizes and simple Au nanopar-ticles, we show that the near field of the metal-coated tipwith a thick (.l) or thin (!l) metal coating can be mod-eled correctly by the self-consistent response of a thindiskto an external field. The tip field is strongly dependenton tip geometry, such as metal-coating thickness and tipcore radius. For example, for tips with a thick metalcoating, the tip field is enhanced beneath the metal coat-ing. The measured NSOM images of a small particlewith a size comparable to the tip core diameter are sensi-tive to features in the tip near field. Consequently, themeasured images also depend strongly on tip size and ge-ometry. Our calculated line scans for the images exhibitthe same strong dependence on the tip-aperture size andcoating thickness. For images made with blunt tips hav-ing a thick coating, our model predicts a central minimumand side maxima (called a triplet structure) in an X scanand a broader central minimum but no side maxima in aY scan, where X and Y axes point along the directions par-allel and perpendicular to the polarization direction of theexternal field that drives the tip. For sharp tips, with acoating thickness much less than half of a wavelength,the line scans show a central minimum but do not exhibitside maxima. The triplet structure is absent when sharptips are used. This structure in the line scans is corre-lated to the structure in the tip-field distribution thatarises because the coating has a finite thickness. Thefeatures of the calculated NSOM images formed with bothblunt and sharp tips are consistent with experiment.18

We show that images calculated with the BB model forthe tip fail to reproduce these essential features. No sidemaxima are seen in the line scans for the BB model be-cause there is no enhancement of the field under themetal screen. Moreover the BB model predicts that thewidths of the central minima in the line scans are almostthe same for X and Y scans in contrast to both the experi-ments and our disk model calculations. The BB modelfails to account for these essential features in the tip nearfield because the BB model cannot account for the effectsof finite coating.

This paper is organized as follows. In Section 2 webriefly describe the NSOM experiment and the key fea-tures of the experimental NSOM images of Aunanoparticles.18 In particular, we distinguish betweenthe NSOM images made with two different types (blunt orsharp) of tips with different metal-coating thicknesses.In Section 3 we outline the theory used to simulate theNSOM images of small-particle systems. We describethe thin-disk model for the tip and calculate the transmit-ted light intensity of the tip–particle system that is col-lected in the far field. In Section 4 we first present thecalculated tip fields. Fields calculated with the thin-diskmodel are compared with those obtained with the BBmodel to determine the effect of a finite metal coating ontip fields. Then we discuss the calculated line scans forthe NSOM images of Au nanoparticles and compare themwith experimental images. We show that the thin-diskmodel reproduces the main features of the measuredNSOM images. We decompose the calculated line scansinto contributions from the tip field, the particle field, andthe interference between these two fields. From this de-

composition we determine that the structure in the linescans is primarily related to the structure in the tip fieldsfor the experiments that we simulate. For comparisonwe also calculate the line scans of the NSOM images byuse of the BB model. Our calculations show that the BBmodel does not describe the NSOM experiments. In Sec-tion 5 we give our conclusions.

2. EXPERIMENTAL RESULTSThe NSOM microscope used in our experiments18 is abeetle-style scanner assembly located inside an ellipsoidalcavity with an Al-coated tip at one cavity focus. The cav-ity allows an efficient, symmetric collection of reflectedlight, simultaneous with transmission. In the experi-ments that we model, all images were recorded in trans-mission after illumination with 488-nm light from the tip.Probe fly-height control was achieved by a feedback loopthat maintained a constant shear-force damping of thetip. Care was taken in the interpretation of the experi-mental images to avoid artifacts due to topographicfeedthrough.19 The samples were Au nanoparticles,approximately 100 nm in size, deposited on a silanizedglass surface. A schematic of the experiment is shownin Fig. 1.

Transmission NSOM images of the Au particle were re-corded with two types of tip having different thicknessesof Al coating. For blunt tips the aperture radius was ;70nm and the metal coating was .300 nm thick. Sharptips had a similar aperture radius but a much thinnercoating. In our experiments, several blunt and sharptips were used. Optical images made with blunt tips (seeFig. 2) show a distinctive triplet structure for verticalscans. In the experiments the polarization of the tip farfield was well defined. However, the polarization wasnot measured. By symmetry the polarization must belinear and either parallel or perpendicular to the scan di-rection defined by the triplet structure. Our calculationsshow that the triplet structure occurs for scans parallel to

Fig. 1. Schematic diagram showing a small particle scanned bya fiber tip in a near-field scanning optical microscope. The diskmodel used to calculate the tip field is also indicated.


the polarization. For Fig. 2 the symmetry argument andour calculations imply that the polarization is in the ver-tical direction. (In this paper the polarization directionwas chosen to be the x direction.) The triplet structurehas a broad central minimum when the tip and the Auparticle are centered and side maxima when the Al coat-ing is over the particle. For y scans the central minimumis significantly broader than in the x scans, and there areno obvious side maxima. The optical images made withsharp tips show only a central minimum with interfer-ence oscillations when the tip and particle are widelyseparated in a y scan but not in an x scan. These fea-tures are the essential, reproducible structures in theNSOM images. A detailed description of these NSOMexperiments is presented elsewhere.18

3. OUTLINE OF THE THEORYTo obtain a realistic description of the optical response ofa small particle to the near field of the NSOM tip, it isnecessary to take into account the interaction betweenthe metal-coated fiber tip and the particle. For theNSOM experiments modeled in this paper the tip isscanned over the particle in an xy plane perpendicular tothe fiber tip axis (z axis) at a tip–particle z separationranging from 10 to 50 nm. The metal coating is Al. The

Fig. 2. Constant-gap transmission NSOM image of a 100-nmAu particle recorded with a tip having a thick metal coating.Vertical and horizontal line scans along the indicated lines aredisplayed.

field inside the metal coating rapidly decays away fromthe surface. Only the end of the tip strongly interactswith the particle and makes a dominant contribution tothe far-field intensity. This allows us to include only thispart of the tip in our theoretical analysis. This approxi-mation is critical for simplifying the numerical calcula-tions and will be justified by comparison of the calculatedline scans and the experimental images. In this paperwe model the metal-coated tip as a thin disk with a smallbut finite height h (see Fig. 1). The interior (core) part ofthe disk is glass and serves as an aperture. The remain-ing part of the tip is the Al coating with a finite outer ra-dius R. We assume that an incident electromagneticfield @E0(r)# with frequency v locally illuminates the coreregion of the tip, namely, the incident field exists only in-side the tip core. We assume uniform illumination of thecore for the results presented here. Other illuminationprofiles in the core and profiles that decay into the metalcladding produce similar results. In the absence of theparticle the tip field is the self-consistent response of thethin disk to the incident field. Our model obviously can-not account for light propagation effects through the fiberto the end of the tip. However, it should reproduce theessential features of the field emitted by the tip and theinteraction between the tip and the particle. Our modelshould account for the effects of a finite coating. This isnot possible with the BB model. As will be shown in Sec-tion 4, the finite coating produces structures in the tipfield due to geometric resonances. These structures arereflected in the NSOM images of Au particles measuredexperimentally. The disk model accounts for this; the BBmodel does not. As we will show, we can straightfor-wardly account for the tip–particle interactions by use ofthe disk model. This is not possible with the BB model.

To calculate the light collected in the far field when asmall particle is illuminated by an NSOM tip, we first cal-culate self-consistently the local field inside the tip, mod-eled as the thin disk, and inside the particle. To this endwe adopt the coupled, discrete dipole approximation.13–15

We divide the tip and particle into N small subvolumes V.The optical response of each subvolume is characterizedby the Clausius–Mossotti (CM) isotropic dipole polariz-ability:

a~v, rj! 5 3e0Ve~v, rj! 2 1

e~v, rj! 1 2, (1)

where e(v, rj) is the relative dielectric constant of the tipor particle material at the grid point rj . The CM polar-izability should work well whenever the width of the sub-volume V is much smaller than either l or the screeninglength in the metal. Analytical, finite-size corrections tothe CM relation can be used when V is more comparableto the length scale for variations in the field.20 The localfields inside the tip and particle are determined from thefollowing algebraic equation:

E~ri! 5 E0~ri! 2 m0v2(jÞi

N

GI ~v, ri , rj! • a~v, rj!E~rj!,

i 5 1, 2,..., N, (2)

r1watitccifipfce

ceicppxsnascp


where GI (v, ri , rj) is the dyadic vacuum Green’s function.By using the thin disk to model the tip, we can limit thenumber of dipoles used to model the tip and yet still havea high density of dipoles in the structure that we use as amodel. In the experiment the Au particle is on a si-lanized surface. Substrate effects can be included by theuse of a Green’s function dressed by surface effects.21,22

However, including the effects of a thick glass substratewith a dielectric constant of .2.25 should not make sig-nificant changes in the NSOM image calculations. Typi-cally we find that the effects of the substrate on transmis-sion are small. For that reason we present results herefor the simpler calculation where the effects of the sub-strate are ignored.

Once the local field at each dipole in the tip and particlehas been found from Eq. (2), we calculate the field radi-ated from the tip and the particle to determine the lightcollected in the far field. In the far-field region the elec-tric field radiated from the tip and particle takes theasymptotic form

Efar~r! 5 m0v2exp~ivr/c0!

4pr (j51

N

a~v, rj!

3 expS 2iv

c0n • rjDMI • E~rj!, (3)

where n 5 r/r is a unit vector and the matrix MI is relatedto the spherical coordinate angles u and f by

equire more than 107 dipoles. Routine simulations with07 dipoles are not feasible. To circumvent this problem,e do our simulations with the dipoles spaced 20 nmpart. Each dipole has an effective polarizability aeff ,hat incorporates finite-size corrections to the CM polar-zability [Eq. (1)]. We use a cubic lattice. Each dipole inhe lattice represents the response of a 20-nm cube. Weould use analytic expressions for the finite-sizeorrections.20 However, these corrections are not explic-tly for a cube. Instead we determine numerically thenite-size corrections for cubes. We adjust aeff for the di-ole used to represent a 20-nm cube to produce the samear-field response to a plane-wave driving field as a 20-nmube modeled by a finely discretized lattice of dipoles,ach with CM polarizability.

A comparison of the near-field intensities of a 20-nm Alube, modeled by a cubic lattice of 1-nm-spaced dipoles,ach having CM polarizability, and a single dipole withts polarizability adjusted to give the same far field as theube is shown in Figs. 3 and 4. We use 1-nm-spaced di-oles because finite-size corrections for 1-nm-spaced di-oles are not important. The driving field is an-polarized plane wave with l 5 488 nm. Figure 3hows the intensity in xy planes that are 15, 20, and 40m from the dipole or from the center of the cube (5, 10,nd 30 nm, respectively, from the cube surface). Figure 4hows the intensity in yz planes. In planes far from theube or the dipole the intensities for the cube and the di-ole are similar. Typically, the dipole-field intensity is

MI 5 F 1 2 sin2 u cos2 f 2sin2 u sin f cos f 2sin u cos u cos f

2sin2 u sin f cos f 1 2 sin2 u sin2 f 2sin u cos u sin f

2sin u cos u cos f 2sin u cos u sin f sin2 uG . (4)

The total power of the transmitted light radiated into thesolid angle of 2puc is given by

P 51

2e0c0E

0

2p

dfE0

uc

duuEfar~r!u2r2 sin u, (5)

where uc is the collection (cone) angle in experiment.To calculate the near field of the tip in the absence of

the particle, we use Eq. (2) with the polarizability of theAu particle set to zero. When the tip field at an observa-tion point ri located outside the tip is calculated, the fullGreen’s function including the near-field part must beused and the summation in Eq. (2) must run over all thegrid points inside the tip.

We simulate NSOM images that were made with tipsthat have an outer radius R that can be as large as 450nm. Simulating just the end of one of these tips as a thindisk using a finely discretized lattice of dipoles is difficult.For example, using dipoles spaced 20 nm apart for thesimulations, '10,000 dipoles are needed to simulate amicrometer-wide disk that is one dipole layer thick.However, the dipole spacing should be smaller than theskin depth ('10 nm) of the metal coating. Simulatingthe same disk with dipoles spaced only 1 nm apart would

more localized in the plane than is the intensity for thecube. More substantial differences between the intensi-ties for the dipole and the cube occur on planes close tothe cube surface. The symmetry of the intensity for thecube clearly differs from the symmetry for the dipole in-tensity for the planes 5 nm from the cube. However,most of the cubic symmetry has disappeared 10 nm fromthe cube surface. In our simulations the tip and particleare separated by 20 nm. We expect the tip–particle in-teraction to be modeled reasonably by dipoles with effec-tive polarizabilities that include the finite-size correc-tions. These dipoles should also provide a reasonablemodel for the interactions between 20-nm cubes on thedisk that are widely spaced. The biggest discrepanciesshould occur for the interaction between adjacent cubes.At this distance the shortest-range near-field effects can-not be completely represented by single dipoles. Weshow that these differences are not critical by our com-parison of calculated line scans and experimental images.

In our simulations, l 5 488 nm. At this wavelengththe bulk dielectric constants for the glass core and alumi-num coating are taken to be 2.25 and 234.5 1 i8.5, re-spectively. Finite-size corrections for the glass core arenot important. For Al the corrections are significant.


Fig. 3. Comparison of the local-field intensities of a 20-nm cube and a single dipole with a polarizability adjusted for finite-size effects.The intensities on xy planes are shown. The driving field is an x-polarized plane wave with l 5 488 nm.


Fig. 4. Comparison of the local-field intensities of a 20-nm cube and a single dipole with a polarizability adjusted for finite-size effects.The intensities on yz planes are shown. The driving field is an x-polarized plane wave with l 5 488 nm.


When the finite-size corrections are included to determineaeff for the dipoles that model the Al coating, aeff /V is 30%greater than a/V for the CM polarizability.

4. NUMERICAL RESULTSA. Calculations of the Tip FieldTo obtain a better understanding of the near-field inter-action between the NSOM tip and the Au particles, firstwe study the tip field in the absence of the particle. Wealso investigate the effect that the finite real metal coat-ing of the tip has on the tip field and compare the calcu-lated results with those obtained with the BB model. Ex-tensions of the BB model that treat the screen as arealistic metal with finite thickness instead of a thin, per-fect conductor have been carried out.23 The more realis-tic models produce near fields similar to the BB fields.None of these models accounts for the finite outer radiusof the tip coating. In this paper we limit our comparisonto the simpler BB model. For all calculations presentedin this paper, we assume that the external field thatdrives the tip core is polarized along the x axis and hasunit amplitude. The x component of the tip field makesthe dominant contribution to the NSOM image formation(see Subsection 4.C), so we focus our discussion on thespatial distribution of this component of the tip field.

In Fig. 5(a) we show the x component of the tip field inthe xy plane 20 nm from the end of the tip. For the thindisk used to model the tip in the calculations shown inFig. 5(a), the disk height is 20 nm, the core radius is 70nm, and the outer tip radius is 170 nm (the metal coatingis 100 nm thick). The disk is represented as a lattice ofdipoles spaced 20 nm apart and adjusted for finite-sizecorrections. For comparison, we plot in Fig. 5(b) the xcomponent of the tip field calculated based on theBBmodel with an aperture radius of 70 nm. Note thatthe tip core is vacuum in the BB model. From a compari-son of Figs. 5(a) and 5(b), we see that the tip field ob-tained in both the disk model and the BB model variesrapidly in space and is highly localized near the aperture.The tip field is a maximum beneath the center of the tip.

The tip fields obtained with the two models differ sig-nificantly. In Fig. 5(a) the field near the tip outer edgeon the axis parallel to the polarization of the externalfield is enhanced owing to the large dielectric mismatchbetween the metal coating and vacuum. Obviously, thisfield enhancement near the tip outer edge is absent in theBB model. The field distribution found from the diskmodel exhibits a small-scale granularity that is absent inthe BB model. This granularity reflects the spacing ofthe dipoles used to represent the disk but does not affectqualitatively the essential features of the tip field.

The field distribution just below the aperture is alsodifferent for the two models. In the BB model the distri-bution is broader along the polarization direction x. Inthe disk model the distribution is broader perpendicularto the polarization. This difference is important and rep-resents a real difference between the BB model, whichtreats the metal coating as an infinite plane, and the diskmodel, which accounts for the finite width of the coating.If the coating were a perfect conductor, then the tangen-tial component of the field would vanish on the surface of

the coating. The dominant field component, which is thex component for an x-polarized driving field, would vanishat the inner and the outer edges on the y axis. Inside theaperture the field distribution for the x component wouldbe narrower along y than along x. This is true for eitherthe BB model or the disk model. In the BB model, Excontinues to have a narrower distribution along y thanalong x in the xy planes below the tip end, as shown inFig. 5(b). At large distances from the tip the BB distri-bution becomes circularly symmetric, as expected for far-field diffraction. In contrast, for the disk model, the dis-tribution is narrower along y only on planes very close tothe tip. Farther from the tip the distribution is broaderalong y. The Ex distribution in the xy plane 20 nm fromthe tip is broader along y than along x, as shown in Fig.5(a). This change in the shape of the Ex distribution forthe disk model results from diffraction at the outer edgeof the coating. For a coating with a finite thickness thetangential field component also vanishes on the outeredge. Thus Ex should be small at the outer edge on the yaxis. This produces diffraction at the outer tip edge,pushing Ex out from the tip edge along the y axis, broad-ening the field distribution along y. This diffraction atthe outer edge can be accounted for only by a model witha finite coating. The symmetry of the field distributionclearly shows the diffraction at the outer edge. As a com-

Fig. 5. Magnitude of the x component of the tip field in the xyplane 20 nm from the end of the tip. (a) Field calculated in thedisk model: disk height, 20 nm; outer tip radius, 170 nm; coreradius, 70 nm. (b) Field calculated in the BB model: apertureradius, 70 nm; wavelength, 488 nm. The tip is centered abovethe origin. The other parameters used in the calculations aregiven in the text.


parison of calculated and experimental line scans for theNSOM images will show, this effect of the diffraction atthe outer edge produces the asymmetry in the centralpeak of the observed images and clearly reveals impor-tant differences between the field distribution for BBmodel tips and the field distributions of real tips.24

In the near-field region the tip field calculated in thedisk model decays rapidly with an increase in distancefrom the tip end. This is consistent with the predictionby the BB model. Our calculations also show that the Eyand Ez obtained for the two different models can be dif-ferent owing to the effects of the finite coating.

To see how the tip field depends on the metal-coatingthickness, we calculate the tip field by using the diskmodel for different outer tip radii with a fixed apertureradius of 70 nm and a disk height of 20 nm. In Figs. 6(a)and 6(b) we show the magnitude of Ex in the xy plane 20nm from the tip end for tips with outer radii of 390 and450 nm, respectively. Comparing Figs. 5 and 6, we seethat the finite metal coating strongly influences the tipfield below the coating, as well as directly below the aper-ture, and should have notable effects on the NSOM im-ages. Our numerical results indicate that the spatial de-pendence of the tip field in the xy plane is stronglydependent on the thickness of the metal coating. In par-ticular, Ex exhibits an obvious wavelike behavior whenthe thickness of the metal coating is larger than l/2. Alocal maximum in the tip field occurs beneath the Al coat-ing. This maximum occurs when the coating is thick

Fig. 6. Magnitude of the x component of the tip field, calculatedin the disk model, in the xy plane 20 nm from the end of the tip.The tip outer radius is (a) 390 nm, (b) 450 nm. The other pa-rameters of the disk model are as in Fig. 5.

enough that the field diffracting away from the apertureand at the tip edge can undergo half-period oscillationsand set up a standing wave, a geometric resonance, belowthe end of the tip. We show below that this structure inthe tip field is revealed in the NSOM images and is re-sponsible for the triplet feature in the NSOM imagesmade with blunt tips.

The BB model fails to predict the field enhancement be-neath the coating and fails to model correctly the width ofthe field distribution beneath the aperture. These re-sults suggest that the BB model fails to account for thestructure in the tip field owing to the finite coating. InSubsection 4.B we show that this structure in the tip fieldcorrelates with the structure in the line scans and thatthe effects of a finite coating must be included correctly toexplain the experimental images.

B. Line Scans of NSOM Images of Au ParticlesWe now present the calculated line scans of the NSOMimages of Au particles. For all the calculations pre-sented in this subsection a rectangular Au particle, 1003 100 3 60 nm (x, y, z), is employed. This particle sizeis very close to that used in our NSOM experiments. Forthe collection cone angle, we take uc 5 30°, which corre-sponds to the experimental numerical aperture. It is dif-ficult to determine experimentally the precise apertureradius and outer tip radius. We analyze the line scans ofthe NSOM images by varying these parameters. In Figs.7 and 8 we show the transmitted light power collected inthe far field as a function of the tip position when the tipis scanned along the X and the Y axis, taking the outer tipradius as a parameter. To compare the calculations withthe corresponding experiments with sharp and blunt tips,small ('170-nm) and large ('390-nm) tip outer radii arechosen for our calculations. Note that the tip and par-ticle are centered when x 5 0 and y 5 0. In the calcula-tions we assume that the tip has a constant separationfrom the top of the particle along the z direction of 20 nm.The tip–core radius is 70 nm, and the disk height is 20nm. At l 5 488 nm the dielectric constant of gold is22.11 1 i4.16. The finite-size corrections for Au weredetermined, as was done for Al, by requiring that the di-pole used to represent a 20-nm Au cube produce the samefar-field response as a 20-nm cube modeled by a finely dis-cretized lattice of dipoles, each with the CM polarizabilityof Au. These finite-size corrections increase the polariz-ability by 10%. As shown in Figs. 7 and 8 the line scansof the NSOM images depend strongly on tip size and onscanning direction. This dependence correlates with thetip-field dependence on tip geometry and scan direction.In particular, we note from Fig. 7 that a central minimumis formed in the optical image for a sharp tip (the outerradius of the tip is small, say 170 nm). When the tip andparticle are centered the transmitted light intensity is aminimum. As the tip is scanned away from the particle,the optical signal gradually increases and approaches thebackground value. A steplike structure appears in the Xscan when the tip edge passes through the particle edge.The X scans show more complicated structures for a blunttip (the tip radius is large, say 390 nm). We see from Fig.7 that the X scan shows both a central minimum and sidemaxima when the metal coating is wider than l/2. In ad-


dition, oscillations due to the interference between the tipfield and the scattered field from the particle can be seenfor blunt tips. For Y scans (see Fig. 8) the results are dif-ferent. For the blunt tip, there is a central minimumthat is broader than for an X scan, but there are no sidemaxima. The shape and widths of the central minimumare consistent with the shape and widths of the tip-fielddistribution predicted with the disk model. Thus ourmodel reproduces the experimentally measured tripletstructure for the blunt tip (a dark spot at the center andtwo bright spots on both sides along the polarization di-rection of the incident field). For the sharp tip the Y scanshows an oscillatory interference behavior when the tip isscanned far away from the particle. This is qualitativelyin agreement with experiment. A detailed comparison of

Fig. 7. X scans for the NSOM images of a 100 nm 3 100 nm3 60 nm (x, y, z) Au particle for different tip sizes: core radius,70 nm; disk height, 20 nm. The outer tip radii used in the cal-culations are indicated. The tip–particle separation in the z di-rection is 20 nm. The particle is centered at the origin.

Fig. 8. Y scans for the NSOM images of a 100 nm 3 100 nm3 60 nm (x, y, z) Au particle for different tip sizes. The otherparameters of the disk model are as in Fig. 7.

theory and experiment shows that we can predict quanti-tatively the width of the central minimum and the posi-tions of the side maxima.18

We also calculated line scans for intermediate-size tipswith outer radii ranging from .250 to 290 nm. Our cal-culations show a striking contrast reversal in the NSOMimages. However, the contrast reversal disappears as weincrease the disk height used in our model calculations.The metal coating is close to l/2 for a tip radius of ;290nm. It is possible that a resonant-excitation condition forthe model tip–particle system is almost satisfied for anintermediate-size tip when the disk is thin. However,the correct description of the NSOM images made withintermediate-size tips may require that we use a diskmodel with a larger height. We do not have experimen-tal data made with intermediate-size tips to clarify thispoint.

To determine whether a change in the core radiusmodifies the NSOM images for sharp and blunt tips, wecompare in Figs. 9 and 10 the line scans along the x axisfor 170-and 390-nm tips with different core radii. Forsharp tips (Fig. 9), changing the core radius does not af-fect significantly the main features of the line scans (thecore radius changes in a small range because the outer tipradius is small for sharp tips). For blunt tips (Fig. 10)the line scans are similar when the core radius is smallerthan 110 nm. As the core radius reaches 190 nm thetriplet structure in the NSOM images disappears becausethe coating becomes too narrow to provide field enhance-ment beneath the coating.

We now compare line scans of NSOM images calculatedbased on the BB and disk models for the tip. Figure 11shows calculated X scans of the NSOM images of an Auparticle in the BB model. In the calculations the tip flyheight is 20 nm and the aperture radius is varied from 30to 190 nm. In the BB model calculations the influence ofthe particle on the tip field is not included. The particleis excited by a tip field that is determined in the absenceof the particle. Thus, in Eq. (2), E0(ri) is the tip field cal-culated by direct integration of the vector potentials fromthe BB model12 and E(ri) is the local field inside the par-ticle. In the far field the BB tip field can be determinedthrough the magnetic dipole radiation, i.e.,

EfarBB~r! 5

4a3v2

3pc02 ~n 3 ey!

exp~ivr/c0!

r, (6)

where a is the aperture radius and ey is the unit vectoralong the y axis. The far field radiated from the particleis still calculated by use of the asymptotic form of theGreen’s function, as in Eq. (3). The total far field is thesum of the BB tip field and the scattered field from theparticle. We see from Fig. 11 that varying the tip-aperture size leads to an obvious broadening of the cen-tral minimum of the line scan and a decrease in the imagecontrast. Otherwise the qualitative structure of the linescan is almost independent of the aperture size. This isin contrast with our experiment and the predictions of thedisk model. In Fig. 12 we compare the X (solid curve)and Y (dotted curve) scans for the NSOM images of theAu particle obtained in the BB model. In the calcula-tions the aperture radius of 70 nm is used. In the BB


model calculation the width of the central minimum inthe X and Y scans of the NSOM images is almost thesame. This is also in contrast with both experiment andour disk model calculation discussed above. This resultsuggests that the BB model does not appropriately de-scribe the tip field of the metal-coated tip used in our ex-periments.

It is also interesting to determine the reaction of the tipto the particle field and the effect of this reaction on theline scans of the NSOM images. Neglecting the reactionof the tip to the particle field is equivalent to assumingthat the tip drives the particle, but the tip does not re-spond to the field radiated from the particle. This reac-tion is included if Eq. (2) is solved self-consistently for alldipoles. This reaction is excluded if the field at a dipolein the tip is determined only by the external driving field

Fig. 9. X scans for the NSOM images of a 100 nm 3 100 nm3 60 nm (x, y, z) Au particle for a fixed outer tip radius of 170nm with different aperture sizes, i.e., 30, 50, and 70 nm. Diskheight, 20 nm; tip–particle separation in the z direction, 20 nm.

Fig. 10. X scans for the NSOM images of a 100 nm 3 100 nm3 60 nm (x, y, z) Au particle for a fixed outer tip radius of 390nm with different aperture sizes, i.e., 30, 70, 110, 130, and 190nm. Disk height, 20 nm; tip–particle separation in the z direc-tion, 20 nm.

plus the fields from the other dipoles in the tip. In Fig.13 we compare the results for line scans along the x direc-tion with (solid curve) and without (dotted curve) includ-ing the reaction of the particle field on the tip in our diskmodel. In these calculations the tip outer radius is 390nm, the aperture radius is 70 nm, and the disk height is20 nm. When the tip and particle are close the effects ofignoring the tip response to the particle field are notice-able but do not affect qualitatively the main features inthe line scans of the NSOM images. When the tip is farfrom the particle (the separation is larger than l), the re-action of the tip on the particle field is negligible. Thisimplies that the important features of the NSOM imagesof a small particle are basically determined by the opticalresponse of the particle to the near field of the tip.

C. Line Scans: Imaging the Tip FieldThe structure in the line scans correlates with the struc-ture in the tip fields calculated with the disk model. Thissuggests that the tip field is being mapped in the trans-mission NSOM images that we simulate. To show this

Fig. 11. X scans for the NSOM images of a 100 nm 3 100 nm3 60 nm (x, y, z) Au particle calculated in the BB model for dif-ferent aperture sizes, i.e., 30, 70, 110, 130, and 190 nm. Tip–particle separation in the z direction, 20 nm.

Fig. 12. X (solid curve) and Y (dotted curve) scans for the NSOMimages of a 100 nm 3 100 nm 3 60 nm (x, y, z) Au particle cal-culated in the BB model. Aperture radius, 70 nm; tip–particleseparation in the z direction, 20 nm.


more clearly, we separate the contributions from the tipfield, from the scattered field of the particle, and from theinterference between the tip and particle fields to thetransmitted intensity. To do so, we consider the case inwhich the influence of the particle field on the tip field isignored. In that case the transmitted light emitted fromthe tip is independent of the particle position. In Fig. 14we show the total transmitted light ( } uE tip

ff 1 E Auff u2),

the transmitted light emitted from the tip ( } uE tipff u2),

and the transmitted light scattered from the particle( } uE Au

ff u2), where E tipff and E Au

ff are the far field emittedby the tip and scattered by the Au particle. The remain-ing contribution to the total transmission, not accountedfor by the tip field or by the particle field, must be due tointerference between the tip and the particle far fields

Fig. 13. X scans for the NSOM images of a 100 nm 3 100 nm3 60 nm (x, y, z) Au particle calculated in the disk model with(solid curve) and without (dotted curve) including the reaction ofthe tip to the particle field. Core radius, 70 nm; outer tip radius,390 nm; disk height, 20 nm; tip–particle separation in the z di-rection, 20 nm.

Fig. 14. Contributions to the X scan of the NSOM image of a100 nm 3 100 nm 3 60 nm (x, y, z) Au particle made by the tipfield, the particle field, and the interference between the tip andparticle fields. The total transmitted intensity is also shown.All intensities are normalized by the constant contribution madeby the tip field. The aperture radius is 70 nm. The tip–particleseparation in the z direction is 20 nm.

( } E tipff

• E Auff* 1 c.c.). The results for a blunt tip are

shown in Fig. 14. Each contribution is normalized by theconstant contribution due to the tip field. The contribu-tion from the particle field is positive and can produceonly bright spots. Thus the intensity from the particlefield cannot explain the central dark minimum in the linescans. The dominant contribution to the central mini-mum must come from interference between the field radi-ated by the tip and the field scattered from the particle.The central feature is dark because the particle field isdriven by the tip field out of phase with the tip field whenthe aperture is centered over the particle. When the par-ticle is not under the aperture the scattered particle fieldis small and the significant contribution to the total inten-sity comes from the interference. Bright and dark spotsoccur in the images as the particle is scanned owing to thevariation in the phase of the tip near field that excites theparticle.

Next we determine which components of the tip fieldmake important contributions to the NSOM images whenthe tip is excited by a linearly polarized incident fieldalong the x axis. In Fig. 15 we compare the calculated Xscans of the NSOM images obtained when the particle isexcited by the total tip field (solid curve) and by only the xcomponent of the tip field (dotted curve). We show re-sults for the blunt tip that were obtained with the diskmodel. From Fig. 15 we see that excitation by Ex pro-duces the important structure in the scan. For Y scansthe difference between the two calculations is evensmaller. The triplet structure and the width of the cen-tral minimum are determined by the x component of thetip field. When only the z component of the tip field isused to excite the particle, the central minimum is muchsmaller than the side maximum. Excitation by the zcomponent alone produces structure in the line scans thatis similar to the small differences between the line scansdue to excitation by the total field and excitation by the xcomponent alone. We conclude that the main structure

Fig. 15. X scans for the NSOM images of a 100 nm 3 100 nm3 60 nm (x, y, z) Au particle calculated in the disk model whenthe particle is excited by the total tip field (solid curve) and byonly the x component of the tip field (dotted curve). Apertureradius, 70 nm; tip–particle separation in the z direction, 20 nm.


of the NSOM images is determined essentially by the op-tical response of the particle to the x component of the tipfield.

Finally, we show that the x component of the tip nearfield is being mapped in the transmission images. Acomparison of the far-field transmission that is due to theinterference between the tip and particle far fields withthe x component of the tip near field at the center of thetop face of the particle, Ex,tip

nf , is shown in Fig. 16 for ascan along the polarization direction with a blunt tip.The tip field and the interference term are strongly corre-lated. In Fig. 16, Ex,tip

nf is scaled to have the same centralpeak as in the interference contribution. The phase ofEx,tip

nf must be shifted by p to match the phase of the in-terference term. Ex,tip

nf shows a sharper structure than

Fig. 16. Comparison of the x component of the tip near field, 20nm from the end of the tip, with the far-field transmission due tothe interference between the tip and particle fields. An X scanmade by a blunt tip with a 390-nm outer radius is shown. Thereal and imaginary parts of the field are scaled to have the samepeak amplitude as the interference term.

Fig. 17. Comparison of the intensity due to the x component ofthe tip near field, 20 nm from the end of the tip, with the far-fieldtransmission due to light scattering from the Au particle. An Xscan by a blunt tip with a 390-nm outer radius is shown. Theintensity is scaled to have the same peak amplitude as light scat-tering from the particle.

does the interference term. The tip field that drives theparticle is an average of the tip field over the entire par-ticle, not just the tip field at the center of the top face.Making the comparison with an averaged tip field shouldreduce the small differences shown in Fig. 16. SinceEAu

ff } Ex,tipnf , the intensity scattered by the particle

should be proportional to uEx,tipnf u2. Figure 17 shows that

the intensity scattered by the particle is strongly corre-lated to uEx,tip

nf u2. The peak of uEx,tipnf u2 is scaled to match

the peak of the scattered intensity. The small differencesbetween the two scans in Fig. 17 are reduced further bycomparing with uEx,tip

nf u2 averaged over the particle ratherthan with just uEx,tip

nf u2 at the center of the top of the par-ticle as was done for Fig. 17.

5. CONCLUSIONWe have presented a theoretical investigation of the near-field interaction between NSOM tips and Au nanopar-ticles and have compared our calculations with experi-ment. We have modeled the metal-coated tip as a thindisk consisting of a glass core and a metal-coating layerand have calculated the tip field by use of the discrete di-pole approximation. The thin disk is a simple model forthe tip. Nonetheless the model can be used to interpretthe experiments. Detailed calculations for thicker disksare still needed for testing the capabilities of this model.The structure in the tip field occurs beneath the metalcoating owing to the finite coating thickness. This struc-ture is not predicted in the commonly used Bethe–Bouwkamp model. The influence of the tip size andshape on the tip field have been studied. Taking into ac-count the mutual interaction between the tip and particle,we have calculated the line scans of the NSOM imagesalong the directions parallel and perpendicular to thelight-polarization direction. We found that the NSOMimages strongly depend on the tip-aperture size and coat-ing thickness. The key features of the experimental im-ages formed with both blunt and sharp tips are repro-duced by our model. For images made with blunt tipsthe model predicts a triplet structure in the X scans and asingle, broader central minimum in the Y scan. We cor-rectly predict the width of the central minimum and theposition of the side maxima measured experimentally.The side maxima in the X scan arise because there is astrong enhancement in the tip field just below the metalcoating along the polarization direction when the coatingis wider than l/2. When the coating thickness is muchless than l/2, as for sharp tips, this field enhancement issuppressed and the side maxima disappear. The widthof the central minimum correlates with the width of thefield distribution beneath the aperture. The shape ofthis distribution is determined by the diffraction aroundthe outer edge of the tip. Images calculated with thesimple Bethe–Bouwkamp model for the tip fail to repro-duce these essential features because the Bethe–Bouwkamp model cannot account for the effects of a finitecoating. No side maxima are seen for this model becausethere is no enhancement of the field under the screen.Moreover the Bethe–Bouwkamp model predicts that the


widths of the central minima are almost the same for Xand Y scans in contrast to both the experiments and ourmodel calculations.

In NSOM the distinction between sample and probe isfuzzy. Our experimental and calculated NSOM imagesdepend on the tip size and cladding. Different tip modelsproduce different images. The calculated images followthe tip field. We conclude that in our experiments theparticle acts as the probe and the tip is the sample, notvice versa. One of the challenges for NSOM is to providethe correct interpretation of measured images. In our ex-periments the NSOM images map the distribution of thetip fields and provide important information about the ef-fects of tip geometry on tip fields and NSOM images.

*Present address, Intel Corporation, 350 East Plume-ria Drive, San Jose, California 95134.

†E-mail, [email protected].

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modeling illumination-mode near-field optical microscopy of au nanoparticles

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