Model for quantitative tip-enhanced spectroscopy and the extraction ofnanoscale-resolved optical constants

Alexander S. McLeod,1, ∗ P. Kelly,1 M. D. Goldflam,1 Z. Gainsforth,2 A. J. Westphal,2

Gerardo Dominguez,3, 1 Mark H. Thiemens,1 Michael M. Fogler,1 and D. N. Basov1

1University of California, San Diego2Space Sciences Laboratory, University of California at Berkeley

3California State University San Marcos(Dated: October 29, 2018)

Near-field infrared spectroscopy by elastic scattering of light from a probe tip resolves opticalcontrasts in materials at dramatically sub-wavelength scales across a broad energy range, withthe demonstrated capacity for chemical identification at the nanoscale. However, current modelsof probe-sample near-field interactions still cannot provide a sufficiently quantitatively interpreta-tion of measured near-field contrasts, especially in the case of materials supporting strong surfacephonons. We present a model of near-field spectroscopy derived from basic principles and verified byfinite-element simulations, demonstrating superb predictive agreement both with tunable quantumcascade laser near-field spectroscopy of SiO2 thin films and with newly presented nanoscale Fouriertransform infrared (nanoFTIR) spectroscopy of crystalline SiC. We discuss the role of probe ge-ometry, field retardation, and surface mode dispersion in shaping the measured near-field response.This treatment enables a route to quantitatively determine nano-resolved optical constants, as wedemonstrate by inverting newly presented nanoFTIR spectra of an SiO2 thin film into the frequencydependent dielectric function of its mid-infrared optical phonon. Our formalism further enables tip-enhanced spectroscopy as a potent diagnostic tool for quantitative nano-scale spectroscopy.

I. INTRODUCTION

Since Synge’s 1928 letter to Einstein proposing a boldmethod for optical imaging beyond the diffraction limit1,sub-wavelength optical characterization techniques haveremained subjects of intensive interest and fierce debateowing to their transformative potential. Among suchtechniques, apertureless near-field scanning optical mi-croscopy (ANSOM)2,3 has shattered the diffraction limit,achieving optical resolutions better than λ/1000 at in-frared and THz frequencies4–6.

Recent coupling of ANSOM to a broadband coherentinfrared light source and asymmetric Michelson inter-ferometer has enabled Fourier transform infrared spec-troscopy at the nanometer length scales (nanoFTIR)7–9,in switchable combination with single-frequency imagingby the pseudo-heterodyne (PSHet) detection scheme10,11.These novel interferometric techniques detect both am-plitude and phase12–14 of the probe-scattered “near-fieldsignal”, which encodes nano-scale near-field optical con-trasts from the sample and transmits them to the far-field. While applications to nanoscale chemical sensingat vibrational “fingerprint” energies are obvious9,15,16,the utility of this instrument for fundamental nano-scale studies of correlated electron systems are equallycompelling17–24.

ANSOM employs a conductive or dielectric AFMprobe as both an intense near-field source and scattererof light into the far-field. The mechanism of optical con-trast has long been understood intuitively via the sim-

∗ asmcleod@physics.ucsd.edu

ple point dipole model25,26, in which radiation scatteredfrom a small polarizable sphere of radius a illuminated byan incident field Einc is modulated through electrostaticinteraction with a material surface located a distance daway in the z-direction:

αeff ≡ Pz /Einc =α

1− αβ/(16π(a+ d)3

) (1)

with α ≡ 4πa3 and β ≡ ε− 1

ε+ 1.

Here α denotes the “bare” polarizability of the sphereproducing a vertical dipole moment Pz, and β denotesthe quasi-static limit of the Fresnel coefficient rp(q, ω).A function of both frequency ω and in-plane momentumq, the Fresnel coefficient describes the relative magnitudeand phase of p-polarized light reflected from the surfaceof material with frequency-dependent dielectric functionε(ω).

While important theoretical advances havebrought near-field spectroscopy beyond qualitativedescriptions27–32, available models describing the probe-sample near-field interaction remain beset by criticallimitations:

1. Many general formulations, although formally ex-act, prove cumbersome to implement for practicalcalculation beyond reduction to the simple pointdipole model33,34. Field retardation and antennaeffects of the probe are explored formally, but notquantitatively.

2. Although the near-field interaction may be de-scribed as an exact scattering problem, many so-lution methods rely on perturbation expansions

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308.

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cond

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.mtr

l-sc

i] 3

0 Ju

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14

2

in powers of the sample response factor β orrp.

9,32,34,35 One can show that such series are di-vergent beyond modest values of rp (the “strongcoupling” regime), leaving this method inapplica-ble for the analysis of crystalline solids and stronglyresonant plasmonic systems.31

3. A number of tunable geometric parameters with adhoc or empirical values are introduced to quantita-tively fit measured data. These include fractionalweights of relevant probe surface charge,30,31 effec-tive probe size and geometry,28–30 the bare tip po-larizability, etc.20 The multitude of ad hoc tunableparameters provides an unreliable recipe for pre-dictive modeling or quantitative interpretation ofdata.

To address these extant shortcomings, the aim of thepresent work is threefold. We first present a new modelof probe-sample near-field interaction, the lightning rodmodel, whose generality allows exploring the influence ofboth probe geometry and electrodynamic effects, whileremaining formally exact in both theory and implemen-tation. Mathematical reduction of this formalism back tothe point dipole model will make clear that unnecessaryad hoc assumptions underpin prevailing models30,31, andthat geometric and electrodynamic considerations mustultimately play a role in their vindication.

Second, we demonstrate this model’s capability to pre-dict spectroscopic near-field contrast in the case of lay-ered structures, which exhibit a strongly momentum-dependent optical response, as well as strongly reso-nant systems, through comparison with near-field spectrameasured on thin films of silicon dioxide (SiO2) and bulksilicon carbide (SiC). Our measurement apparatus is anovel infrared near-field microscope equipped for bothPSHet imaging and broadband nanoFTIR spectroscopy,described in Appendix A.

Finally, we present a method to invert the lightning rodmodel to extract a material’s complex dielectric functionwith nanoscale resolution, which we demonstrate explic-itly for an SiO2 thin film sample. This procedure, com-bined with the unifying formalism of the lightning rodmodel, provides a powerful diagnostic tool for quantita-tively studying the nano-resolved optical properties ofmolecular systems, phase-separated materials, and con-fined nanostructures using ANSOM.17

II. THE LIGHTNING ROD MODEL

Our model describes the near-field interaction betweenan ANSOM probe and a sample surface through a generalformalism that is in principle exact, without appealing toempirical or ad hoc parameters. The chief observable ofANSOM is the radiation field of a polarized probe placedin proximity to a sample (experimental details describedin Appendix A). Since the field originates from reorgani-zation of charges developing on the probe surface in re-

sponse to an incident illumination field, together with thenear-field of the proximate sample, we begin by formingan expression for this instantaneous charge distribution.

Constraining our attention to nearly axisymmetricprobe geometries, the charge distribution is succinctlyexpressed through a linear charge density λQ(z) ≡dQ/dz(z), Q denoting charge and z the probe’s axialcoordinate. In the quasi-static approximation, the fieldErad re-radiated or back-scattered from the probe is pro-portional to its induced dipole moment

Pz =

∫dz z λQ(z). (2)

Appendix E presents how the radiated field is obtainedfrom λQ(z) when electrodynamic phenomena are of fun-damental importance, i.e. when the size of the scattereris comparable to the light wavelength. This relationshiphighlights the central role of the induced charge distri-bution in determining the measured observables of near-field spectroscopy.λQ(z) can be written as the sum of charges induced

by the incident field and those differential contributionsdλQ nf(z) induced by reflection of probe-generated near-fields off the proximate sample:

λQ(z) = Einc Λ0(z) +

∫dλQ nf(z). (3)

Here Λ0(z) denotes the induced charge per unit field re-sulting from incident illumination. Its functional formdepends on the nature of the incident field and the probegeometry, but its contribution to λQ(z) scales with themagnitude of the incident field Einc. The induced chargeelements dλQ nf(z) take the form:

dλQ nf(z) = dQ

∫∫ ∞0

dq dq′ G(q) Γt→s(q)

×R(q, q′) Γs→t(q′) Λ(q′, z). (4)

Here q and q′ denote in-sample-plane momenta forFourier components of the near-field reflected by thesample in response to the polarized probe. Provided aplanar sample geometry, this parameterization offers asparse basis in which to efficiently solve the problem, incontrast with real-space treatments (e.g. the finite ele-ment method). Eq. (4) can be understood through thephysical mechanism shown schematically in Fig. 1b anddescribed as follows in terms of field emission from theprobe and sample reflection.

Charge elements dQ = dz′λQ(z′) on the probe formrings with radii Rz′ along its surface. Considered in theangular spectrum decomposition (Appendix B), each ringemits a distribution of axisymmetric p-polarized evanes-cent fields whose Fourier components are weighted byG(q) = q J0(qRz′). Ji(. . .) is the Bessel function of thefirst kind at order i, with cylindrical coordinates ρ and z.These emitted fields (so-called evanescent Bessel beams)reach the sample surface a distance d below the tip apex

3

5 µm 250 nm

0"

Eq!

Λ(q, z)"[ norm.]

a) b) c)

0"

-1"

-½"

1" 0" -1"2"

-2"

+½"

FIG. 1. a) Scanning electron micrograph of a typical commercial near-field probe exhibiting a conical geometric profile andcharacteristic length scales (probe length and tip radius) separated by nearly three orders of magnitude. The surface profile(blue) is considered in Sec. VI. b) Schematic description of the probe-sample near-field interaction, involving emission ofcylindrical evanescent fields from charge elements in the probe and their reflection by the sample. c) Probe response function

Λ(q, z) (defined in the main text) computed by the boundary element method (Appendix C) for evanescent fields ~Eq of increasingmomenta q. Dashed curves indicate the geometric profile of the probe, and surface charge distribution profiles are normalizedby their minimum values for viewing purposes.

(z = 0) via propagator Γt→s(q) = e−q(z′+d) and in the

empty tip-sample gap −d < z < 0 take the divergence-free form (per unit charge):

~Eq(~r) = G(q) (J0(qρ) z + J1(qρ) ρ) eq(z−z′). (5)

In general, the sample may subsequently scatterevanescent fields with momentum q into distinct Fouriercomponents q′ as described through a differential sampleresponse function R(q, q′). For samples with continu-ous in-plane translational symmetry (e.g. flat surfaces,layered structures) this response function reduces to theFresnel reflection coefficient for p-polarized light,

R(q, q′) = rp(q) δ(q − q′), (6)

written here as a function of the in-plane momentum q ofincident light, with δ(. . .) denoting the Dirac delta dis-tribution. These scattered fields extend from the sam-ple surface via propagator Γs→t(q

′) = e−q′(z+d) and

re-polarize the probe, inducing a linear charge density(per unit field) described by a probe response functionΛ(q′, z). This formalism accommodates the non-trivialinfluence of realistic tip geometries on the functional formof illumination- and sample-induced charge distributionsΛ0(z) and Λ(q′, z).

Induced charge densities can be pre-computed for anaxisymmetric probe of arbitrary geometry using a simpleboundary element method (Appendices C and D). Fig.1c displays Λ(q, z) for several values of q, computed forthe case of a hyperboloidal (conical with rounded tip)probe geometry with tip curvature radius a. As shownin Fig. 1c, the density of charge dramatically accumulat-ing at the probe apex - the celebrated lightning rod effect- increases roughly as 1/q. This results from the req-uisite screening of evanescent fields by induced charges

distributing a distance δz = 1/q along the probe surface.The momentum-dependent lightning rod effect is criti-cally absent from models of the probe-sample near-fieldinteraction lacking a faithful geometric description.

Confining our attention to planar sample geometries,Eqs. (3), (4), and (6) together describe a self-consistentquasi-one-dimensional scattering process:

λQ(z) = Einc Λ0(z)−∫ ∞

0

dq λQ(q) · q e−2qd rp(q) Λ(q, z)

(7)

with λQ(q) ≡∫ L

0

dz′λQ(z′) · e−qz′ J0(qRz′). (8)

The integral transform in Eq. (8) denotes summation ofnear-fields emitted from charges along the entire length ofthe probe, 0 < z < L. A similar integral transformationz → s applied to λ(z) and Λ(q, z) in Eq. (7) yields an

integral equation in λ(s):

λQ(s) = Einc Λ0(s)−∫ ∞

0

dq λQ(q) · q e−2qd rp(q) Λ(q, s).

(9)Eq. (9) resembles the Lippman-Schwinger equation of

scattering theory36. Here Λ0(s) and Λ(q, s) play the roleof in- and out-going “scattering states.” Tractability inthis scattering formalism is afforded by the axisymmet-ric approximation, which preserves most fundamental as-pects of the system geometry.

Provided knowledge of the probe response functionsΛ0(z) and Λ(q′, z), Eq. (9) is soluble by traditionalmethods37 after discretizing q to a set of Gauss-Legendrenodes qi. We found evaluation at Nq ≈ 200 such nodessufficient for numerical accuracy to within 1%. Only afinite range of momenta 0 ≤ q ≤ qmax need be considered

4

in practice, since Λ0(s) and Λ(q, s) drop precipitously inmagnitude above a cutoff momentum s ∼ 1/a, with athe smallest length scale relevant to the probe geometry,in this case the radius of curvature at the probe apex,a ≈ 30 nm for many commercial probe tips. This reflectsthe inability of strongly confined fields (e.g. q ∼ nm−1)to efficiently polarize the probe.

The solution to Eq. (9) is then

~λQ =~Λ0

I−ΛGEinc, (10)

where the denominator is taken in the sense of matrix in-version, vectors imply functional evaluation at momentaqi, I is the identity operator, and other matrices aredefined as

Λij ≡ Λ(qj , si) and Gij ≡ −qi e−2qid rp(qi) δqi δij .(11)

Here δqi is the measure of qi and δij denotes the Kro-necker delta function. Defining similarly a vector of

charge distribution functions[~Λ(z)

]i≡ Λ(qi, z) together

with their associated contributions to the radiated field~erad ≡ Erad

[~Λ]

(see Appendix E), the total inducedcharge and consequent back-scattered field are providedthrough Eq. (7) as:

λQ(z) /Einc = Λ0(z) + ~Λ(z) ·G~Λ0

I−ΛG(12)

Erad /Einc = erad, 0 + ~erad ·G~Λ0

I−ΛG. (13)

Note that dependence on both the tip-sample distance dand the local optical properties of the sample enter theseexpressions through G, whereas geometric properties ofthe probe enter separately through Λ. When applied toa realistic probe geometry, these expressions constitutethe lightning rod model of probe-sample interaction, sonamed for its quantitative description of the strong elec-tric fields localized by an elongated geometry to a pointedapex. The product of Λ and G signifies that strong near-fields from the probe multiplicatively enhance optical in-teractions with the sample surface. Expanding the Eq.(13) inverse matrix as a geometric series reveals an infi-nite sequence of probe polarization and sample reflectionevents, equivalent to the perturbation expansions pre-sented elsewhere9,32,34,35.

Eq. (13) can also recover the point dipole model (Eq.(1)). After simplifying the probe geometry to a metallicsphere of radius a and assuming that all center-evaluated(z = a) fields polarize like the homogeneous incident fieldEinc, we have:

Λ(q, z) = 3/2 (z − a) e−qa, (14)

Λ(q, s) = −a3 s e−(s+q)a, (15)

and pz(q) =

∫ 2a

0

dz z Λ(q, z) = a3 e−qa. (16)

Λ(q, s) is obtained from semicircular Rz in Eq. (8), andexhibits a characteristic maximum followed by a sharp

decay in magnitude near s ∼ a−1. In this case, Eq. (9)

yields λQ(s) in closed form owing to the separability of

Λ(q, s):

λQ(s) =−a3 s e−sa

1− a3∫∞

0dq q2 e−2q(d+a) rp(q)

. (17)

The sphere’s polarization is obtained through Eqs. (7)and (2) as

αeff ≡ Pz/Einc =a3

1− a3∫∞

0dq q2 e−2q(d+a) rp(q)

. (18)

If the sample material is weakly dispersive for q ω/c,rp(q) ≈ β and Eq. (1) is recovered.

Such simplifications are instructive, but this workmakes full implementation of Eq. (13) without recourseto approximation, thus revealing aspects of the probe-sample near-field interaction unresolved by the pointdipole model. While perturbative expansions and thepoint-dipole model may be attractive for their relativesimplicity, they are certainly not expected to be accu-rate. In particular, for large β, nothing short of the fullnumerical solution to Eq. (11) is acceptable for predict-ing experimental observables with quantitative reliabil-ity. Our procedure for doing so is detailed in the followingsections.

The near-field experiments presented in this work uti-lize lock-in detection of the probe’s back-scattered field atharmonics n of the probe tapping frequency Ω to suppressnoise and unwanted background. Simulating this tech-nique, the probe’s back-scattered field Erad (Eq. (13)) isconnected to experimentally observed amplitude Sn andphase φn signals through a sine transform under sinu-soidally varying tip-sample distance d:

sn(ω) = I(ω)

∫ π/Ω

−π/Ωdt sin (nΩt) Erad

(d, rp(q, ω)

)(19)

with d = A (1 + sin (Ωt)) ,

Sn(ω) ≡ |sn(ω)| , and φn ≡ arg sn(ω) . (20)

Here Ω and A are the tapping frequency and amplitude ofthe near-field probe, respectively, and I(ω) denotes thefrequency-dependent instrumental response of the collec-tion optics, interferometer and detector used for the mea-surement. This factor can be removed by normalizing ex-perimental sn(ω) to “reference” near-field signal values,as collected from a uniformly reflective sample materialsuch as gold or undoped silicon. This normalization pro-cess is further discussed in Sec. IV.

A prediction of near-field contrast using the lightningrod model therefore requires calculation of Eq. (13) atseveral values of d; in practice we find 20 such values suf-ficient, with evaluation of Eq. (13) for each requiring sev-eral milliseconds on a single 2.7 GHz processor. Cumu-latively, the calculation remains both realistic and fast,more so than previously reported semi-analytic solutions

5

for realistic probe geometries.38,39 For example, typicalcalculations of a demodulated and normalized near-fieldspectrum across 100 distinct frequencies require less than10 seconds of computation time.

We conclude this formal introduction with a concep-tual clarification. Although the geometry and materialcomposition of the near-field probe implicitly determineits response function Λ(q, z), the formalism embodied byEq. (13) is general and outwardly irrespective of specificproperties of the probe. Also, while plasmonic enhance-ment may be encompassed in Λ(q, z), it is not a prereq-uisite for effective near-field enhancement at the probeapex. Near-field enhancement is attainable through acombination of three distinct mechanisms:40

1. the lightning rod effect proper, due to accumulationof charge at geometric singularities, an essentiallyelectrostatic effect,

2. plasmonic enhancement, due to the correlated mo-tion of surface charges near the plasma frequencyof metals,

3. and antenna resonances, in which the size of anoptical antenna correlates with the incident wave-length in a resonant fashion, a purely electrody-namic effect.

The quasi-static boundary element utilized in this work(Appendix C) reproduces the first of these mechanismsby way of Λ(q, z), whereas its electrodynamic counterpart(Appendix D) reproduces all three. Although plasmonicenhancements are scarcely attainable in metallic probesat infrared frequencies, Secs. III and IV establish theimportant influence of both the lightning rod effect andantenna resonances in near-field spectroscopy.

III. THE QUASI-STATIC CASE

We now apply Eq. (13) to realistic probe geome-tries in the quasi-static limit to investigate whether thequasi-static approximation is appropriate for quantita-tive prediction of near-field contrasts. By reducing thephysical system to electrostatics, this approximation isstrictly justified only in treating light-matter interactionsat length scales much smaller than the wavelength oflight, whereas a typical near-field probe consists of anAFM tip tens of microns in height (Fig. 1a), compara-ble to typical wavelengths encountered in infrared near-field spectroscopy. Consequently, for the assumptions ofa quasi-static probe-sample interaction to remain valid,the emergent behavior of a realistic near-field probe mustbe shown nearly equivalent to those of a deeply sub-wavelength one.

To test this assumption, we consider the near-field in-teraction between a metallic ellipsoidal probe orientedvertically over a planar sample of 300 nm of thermal sil-icon dioxide (SiO2) on silicon substrate. This samplesystem and model probe geometry were considered in a

L

a SiO2 phonon!

FIG. 2. Spectral near-field contrast between the 1130 cm−1

surface phonon polariton of SiO2 and silicon (providing nor-malization) as predicted by the lightning rod model for anellipsoidal probe in the quasi-static approximation. Contrastincreases monotonically beyond experimentally observed lev-els as the probe length is increased.

previous work38, demonstrating the capacity of near-fieldspectroscopy to resolve the ω ≈ 1130 cm−1 surface opti-cal phonon of thermal oxide films as thin as 2 nm. Weextend the theoretical study presented therein to inves-tigate the influence of the probe length L on the ampli-tude of experimentally measurable back-scattered near-field signal S3(ω) (normalized to silicon) predicted by thelightning rod model. The outcome is presented in Fig. 2.

The probe tapping amplitude is 80 nm in these simu-lated experiments, and the radius of curvature a at theprobe apex (equal to the inverse surface concavity) is heldconstant at 30 nm, typical of experiments with commer-cially available near-field probes. The minimum probe-sample distance is taken as d = 0 nm throughout (viz.physical contact, consistent with the established descrip-tion of tapping mode AFM). We describe the thin-filmoptical response with a momentum-dependent Fresnelcoefficient41 (further discussed in Sec. V) using opticalconstants of thermal oxide taken from literature42.

The probe response function Λ(q, z) is computed in thequasi-static approximation once for each probe geometryaccording to a simple boundary element method. Math-ematical details are provided in Appendix C. Whereasin this work we present calculations only for ideally con-ducting metallic probes, Appendix C presents also thegeneral formulation suitable for application to dielec-tric probes. Consequently, the case of a dielectric probepresents no formal difficulty for the model presented here.However, previously reported models present a suitablysimpler description of the “weak coupling regime” inwhich externally excited near-fields may be mapped non-perturbatively13,43,44. We identify this as the regime inwhich a perturbation expansion of Eq. (13) is found to

6

L θ

Einc Esca

0"

1"|Esca / Einc| λ = 10 µm

Einc

b) a)

FIG. 3. a) Scattered field of a realistic near-field probe geometry under plane wave illumination (incident along the viewingangle) as computed by the finite-element method. Oscillatory fields near the tip apex are associated with standing wave-likesurface charge densities resulting from field retardation. b) Lower panel: Field enhancement at the tip apex computedquasi-statically (QS) and electrodynamically (ED) for two probe geometries of varying size illuminated perpendicular to theirprinciple axes. Upper panel: Near-field S3 contrast between SiO2 at the surface optical phonon resonance (ω SO) and siliconsimulated by the fully retarded lightning rod model. The vertical dashed line indicates the length of a typical near-field probe,L ≈ 19µm.

converge, and several terms therein might be summed fora sufficient estimate of near-field scattering.

As shown in Fig. 2, the most dramatic feature of ourquasi-static calculations is the strong variation in normal-ized scattering amplitude with increasing probe length atthe probe-sample polariton resonance. The implication isworrisome: there is no clear rational choice for “effectiveprobe length” when computing the strength of probe-sample near-field interaction in the quasi-static approxi-mation. With a free-space wavelength of light λ ∼ 10 µm,although the largest credibly quasi-static probe lengthL ∼ λ/10 (or L ∼ 20a) provides reasonable qualitativeagreement with data acquired by nanoFTIR under identi-cal experimental conditions (Fig. 7), quantitative agree-ment is clearly only attainable a posteriori, for exampleby fitting to agreeable values of L. Furthermore, theextreme dependence on probe geometry exhibited herediscredits the utility of quantitative “fits” to experimen-tal data. The ill-posed description of near-field couplingafforded by this quasi-static treatment lacks clear predic-tive power. We are therefore compelled towards a con-sistent electrodynamic treatment, which as we will showprovides an unambiguous description of near-field inter-actions with wavelength-scale probes – a case applicableto the vast majority of near-field experiments at infraredand THz frequencies.

IV. THE ELECTRODYNAMIC CASE:NEAR-FIELD PROBE AS ANTENNA

Near-field microscopy is occasionally described as anantenna-based technique, in which the antenna-like near-field probe efficiently converts incident light into stronglyconfined fields at the probe-sample feed-gap17,45–47. Theantenna’s scattering cross section is consequently modu-lated through strong interactions with the sample sur-face to provide the nano-resolved optical contrasts ofANSOM4,48. At a formal level, these considerations leavethe mathematical form of the lightning rod model unal-tered; nevertheless, the probe response function Λ(q, z)must encapsulate the probe’s role as an antenna, particu-larly in the probe’s response Λ0(z) to illumination fields.

As for any antenna, due to retardation and radiativeeffects, the field scattered by a near-field probe is mani-festly dependent on both its size relative to the free spacelight wavelength as well as its geometric profile relativeto the incident light polarization. Such electrodynamiceffects have been demonstrated experimentally47,49. Tocharacterize how these length scales influence the observ-ables of near-field spectroscopy, the full electrodynamicresponses of two probe geometries were computed numer-ically as a function of their overall length L relative tothe free-space wavelength of incident light.

7

A fully retarded boundary element method taking ac-count of field retardation and radiative forcing (math-ematical details provided in Appendix D) was used tocalculate charge distributions Λ0(z) induced on metallicellipsoidal and hyperboloidal probe geometries by inci-dent 10 µm wavelength light (ω = 1000 cm−1). We con-sider here the hyperboloidal geometry to faithfully reflectthe cone-like structure of conventional near-field probeswhich exhibit a taper angle θ ≈ 20 relative to theiraxis in our experiments. A similar hyperboloid probe ge-ometry was applied previously by Behr and Raschke toexplore plasmonic field enhancements39. However, theirfully analytic treatment necessitates a semi-infinite probegeometry treated in the quasi-static approximation, re-quiring an unconventional field normalization method toobtain finite values for the probe response. Their for-malism also left back-scattering from the ANSOM probeunexplored. For our examination, we explore the explicitelectrodynamics of probes with lengths between L = 60nm (rendering a sphere in the ellipsoidal case) and 30 µm,with the apex curvature radius held constant at a = 30nm.

The axisymmetry favored by the lightning rod modelwas maintained throughout these calculations by ap-proximating plane wave illumination by an inwards-propagating cylindrical field bearing a local phase veloc-ity angled towards the tip apex at 60 from the probeaxis (see Appendix D). Validity of this axisymmetric ap-proximation was confirmed through comparison of resul-tant surface charge density profiles with those predictedby full finite-element simulations (Comsol Multiphysics),consisting of a realistic metallic probe geometry (θ = 20

and L = 19 µm) including AFM cantilever, subject toplane-wave illumination. Differences in charge densitywere found to be negligible within microns of the tipapex, suggesting the robustness of key near-field param-eters to fine details of the extended probe geometry. Fig.3a displays finite-element predictions for the magnitude

of the probe’s scattered field ~Esca illustrating the charac-teristically standing wave-like pattern of charge densityalong the probe’s conical surface, a consequence of fieldretardation.

The resultant field enhancement at the probe apexin the absence of a sample calculated by our fully re-tarded boundary element method is shown in Fig. 3b(lower panel) in comparison with the quasi-static case,demonstrating several key phenomena: First, quasi-static probe geometries exhibit field enhancements thatincrease monotonically with the geometric “sharpness”L/a due to the electrostatic lightning rod effect, origi-nating the divergent quasi-static near-field contrast dis-played in Fig. 2. Second, at lengths L = mλ/2for odd integers m ≥ 1, the electrodynamic ellipsoidalprobe exhibits resonant enhancement, whereas minimaare observed for m even. These features signify antennamodes with antisymmetric and symmetric surface chargedensities50, respectively, such as those experimentallycharacterized among similarly elongated near-field probe

geometries47. Due to the axially polarized incident field,Resonant enhancement modes of the hyperboloidal probeare less pronounced and more complicated in character;we discuss them here in no further detail. Finally, it isclear that quasi-static predictions depart from their elec-trodynamic counterparts near a probe length L ∼ λ/10,precisely where quasi-static approximations might be ex-pected to falter lacking the antenna enhancement mecha-nism. Field retardation halts subsequent increases in en-hancement from the quasi-static lightning rod effect, con-ferring a practical limit to realistically attainable near-field enhancements outside the plasmonic regime.

Similarly, the onset of antenna modes is expectedto modulate the intensity of frequency-dependent back-scattered radiation from wavelength-scale near-fieldprobes, opening the possibility to optimally enhance ab-solute near-field signals through application-driven de-sign of novel probe geometries. However, the need for abroadband and normalizable probe response is equallycrucial for spectroscopy applications51. A typical in-frared near-field spectroscopy experiment involves nor-malizing acquired signals to a reference material thatexhibits a nominally flat optical response (e.g. goldor undoped silicon) in order to remove the influenceof instrumental sensitivities9,20,38, including the probe’sfrequency-dependent antenna response. Normalizabilityof this response is typically assumed, but we predict herefor the first time the breakdown of this assumption in thevicinity of strong antenna resonances.

Fig. 3b (upper panel) displays the result of fully elec-trodynamic lightning rod model predictions for the near-field signal S3 SO obtained at the peak probe-sample reso-nance frequency (ω ≈ 1130 cm−1) induced by the strongSiO2 surface optical phonon, normalized to the signalfrom silicon. Whereas an increase in absolute back-scattered signal is expected near the onset of a (radiative)antenna mode, this evidently accompanies a remarkabledecrease in relative material contrast. The effect re-sults from strong cross-talk between the implicit proberesponse coincident with that of a resonant sample.

The explanation becomes clear when considering thatan antenna’s resonance can be strongly detuned by itsdielectric environment.52,53 The point dipole model (Eq.(1)) admits interpretation as a dipole interacting with itsmirror image projected from the sample surface. Extend-ing this interpretation to an antenna, the electrodynamicsystem consists then not of a single antenna, but of a cou-pled antenna-mirror pair, and it is well established thatcoupling an antenna with an exact mirror copy inducesa resonance red-shift54,55. Whereas the mirror couplingscales with the inverse dimer gap size in the case of physi-cal antenna pairs, this coupling scales with rp in our case,and could be appropriately considered a case of dielectricloading.44,52

Consequently, an SiO2 film is expected to detune an-tenna resonances more strongly at ωSO than a Si sub-strate, rendering their respective probe back-scatteringsignals potentially incomparable even when collected at

8

22 nm!SiO2!

Silicon!2 nm! SiO2!

SiO2 Si

Esca Einc

t

a) b)

min.! max.!S3

FIG. 4. a) Near-field response of SiO2 thin films etched to varying thicknesses on a silicon substrate measured by tunableQCL spectroscopy and normalized to silicon38 (see text). The faint curves are provided as guides to the eye. Inset: Samplenear-field signal S3 at ω = 1130 cm−1 overlaid on simultaneously acquired AFM topography. b) Near-field S3 spectra predictedby the lightning rod model using optical constants from literature42. Data points from the 300 nm film are superimposed forpoint of comparison. Our model captures the key features of the data; we infer that discrepancies with ultra-thin film dataresult from substantial variations in optical properties.

the same frequency, since there is no clear way to normal-ize out the effect. Stated another way, interaction witha resonant sample can not only enhance the strengthof a probe’s antenna mode, it can modify the antennabehavior outright, driving the probe towards a regimeof destructive radiative interference. Normalized S3 sig-nals calculated for the electrodynamic ellipsoid (Fig. 3a,solid blue line) therefore resemble a quotient of two res-onance functions, oscillatory but shifted versus the lightfrequency relative to one another. For this extremecase, we might conclude that fluctuations observed in thefrequency-dependent near-field signal radiated from theprobe could associate more with variable dielectric load-ing of the antenna response than with genuine near-fieldcontrast.

Antenna detuning is considerably moderated in thecase of the hyperboloidal probe, whose normalized near-field response at frequencies λ < L exhibits much weakerdependence on the probe length (or, complementarily, onprobing frequency). The normalization procedure there-fore appears sufficient for systematic removal of the probesensitivity at the 20% level in the absence of strong an-tenna resonances. Furthermore, given the clear asymp-totic character of near-field contrast for the broadbandhyperboloidal probe, it would appear acceptable to quan-titatively model normalized near-field signals from sucha probe geometry using electrodynamic charge distri-butions Λ(q, z) computed only for a single characteris-tic frequency. In the case of weak antenna resonances,this renders implementation of the fully retarded light-ning rod model no more complex than the quasi-staticversion. Therefore, all following calculations presented

in this work are electrodynamic and calculated in thisfashion unless otherwise indicated.

Nevertheless, this examination tells a cautionary taleconcerning the use of strongly resonant probes47 forquantitative near-field spectroscopy, wherein convolutionof the probe’s antenna response may not be easily re-moved. However, the resonant enhancement of back-scattered fields by L ∼ λ/2 probes can provide a for-tunate trade-off, with encouraging applications to reso-nantly enhanced THz near-field imaging experiments.

V. MOMENTUM-DEPENDENTLIGHT-MATTER COUPLING

To test the lightning rod model description of systemsexhibiting explicit momentum-dependent light-mattercoupling, we consider a thin film of phonon-resonant SiO2

on silicon substrate. The film thickness t introduces acharacteristic length scale to the sample geometry, asso-ciated with a characteristic crossover momentum q ∼ t−1.Incident evanescent fields exceeding this momentum arereflected much as though bulk SiO2 were present, whereaslower momentum fields can penetrate the film and reflectfrom the substrate38. With the lightning rod model weconsider this momentum dependence exactly and directlycompare its predictions to near-field spectroscopy mea-surements performed using the experimental apparatusdescribed in Appendix A.

Mid-infrared near-field images of SiO2 thin films ofvarying thickness were acquired with a tunable QCL at aprobe tapping amplitude of 50 nm, taking signal from the

9

q-vector

1000

1100

1200

1300

ω[c

m−

1 ]

≤ .25

3

|rp |

10−2 10−1 100 101

q [a−1]

0.0

0.5

1.0

δE(q

)

!SO"

!LO"

!TO"

t = 100 nm a)

b)

q = t -1"

FIG. 5. a) Example of strong surface optical phonon dis-persion for a 100 nm thick SiO2 film on silicon computedby the Fresnel reflection coefficient rp(q, ω) (Eq. (21)). b)The momentum-dependent distribution of electric fields atthe sample surface (dashed line) calculated by the lightningrod model at the tip-sample phonon resonance for a conicaltip in full contact.

underlying silicon substrate for normalization (Fig. 4a).These data were first presented in an earlier work38. Con-trolled film thicknesses were produced through selectiveetching (NT-MDT Co.) and confirmed by AFM heightmeasurements acquired simultaneously with the collec-tion of near-field images. Spectroscopy was obtainedfrom area-averaged near-field contrast levels.

Momentum-dependent Fresnel reflection coefficientswere used to describe these systems41 and provided tothe lightning rod model in order to predict spectroscopicnear-field contrast:

rp(q, ω) =ρ1 + ρ2 e

2ikz,1 t

1 + ρ1ρ2 e2ikz,1 t(21)

with ρi ≡εi kz,i−1 − εi−1 kz,iεi kz,i−1 + εi−1 kz,i

,

and kz,i ≡√εi (ω/c)2 − q2.

Here numeric subscripts 0, 1, 2 correspond with air,SiO2, and silicon, respectively, εi denotes the complexfrequency-dependent dielectric function of the relevantmaterial (ellipsometric optical constants for thermal ox-ide taken from literature42), and t denotes the oxide filmthickness.

Lightning rod model predictions are presented in Fig.4b for comparison with measured data. Agreement is su-perior to that of the simple dipole model and at least

as good as earlier quasi-static predictions with an ad-hoc probe geometry38. In contrast to the prediction ofa blue-shifting phonon resonance with decreasing filmthickness (originating entirely in the Fresnel formulaEq. 21), experimental data indicate a slight red-shiftamong ultra-thin films. This discrepancy should not becounted against our model: although identical opticalconstants were employed for predictions at all film thick-nesses, a growing body of experimental and ab initioevidence suggests legitimate phonon confinement effectscan modify the intrinsic optical properties of nanostruc-tured samples56. Clear discernment of these effects bynear-field spectroscopy opens the possibility for quanti-tatively evaluating the optical properties of nanostruc-tures that exhibit and utilize bona fide three-dimensionalconfinement57.

A clear physical description of the depth sensitivityexhibited in Fig. 4 proves just as valuable as quanti-tative agreement. The onset of a dramatic decrease innear-field signal at the phonon resonance near t ∼ a canbe understood on the basis of the momentum decompo-sition of electric fields emitted by the near-field probe.A straightforward analysis building on Eq. (12) revealsthe following decomposition for probe-generated electricfields by their momenta in the plane of the sample (thebasis given by Eq. (5)):

δE(qi) /Einc =

[Γt→s ΛG

~Λ0

I−ΛG

]i

δq (22)

with[Γt→s

]ij≡ e−qid δij ,

where d is the tip-sample distance and δE(q)/δq is un-derstood in the sense of a distribution function.

Fig. 5b displays δE(q) calculated on resonance withthe SiO2 phonon in comparison with the example dis-persion of a 100 nm SiO2 film on silicon shown in Fig.5a. The surface optical phonon is evident at ωSO,characteristically centered in the Restrahlen band be-tween the transverse optical (ωTO) and longitudinal op-tical (ωLO) phonon frequencies. Given that our SiO2

forms an amorphous layer, indication of these phonon fre-quencies is approximate. Nanoscale thickness introducesconsiderable momentum dependence in the regime rele-vant to probe-sample near-field interactions (q ∼ a−1),effecting a strong phonon response only for momentaq > t−1 as mentioned earlier. The spectroscopic char-acter of the probe-sample near-field response can there-fore be inferred from the momentum-space integral ofδE(q) × rp(q, ω). Note however the explicit rp- and d-dependence of δE(q) by way of G in Eq. (22) amountsto a near-field response strongly super-linear in the sam-ple’s intrinsic surface response.

10

Au SiC

Esca Einc

Au SiC

890"cm(1"

a) b)

FIG. 6. a) Amplitude S3 and phase φ3 of the back-scattered near-field signal from a 6H SiC crystal, measured in the vicinityof the surface optical phonon and referenced to a surface-deposited gold film, as obtained in a single acquisition by nanoFTIR.Left inset: Visible light image (width 60 µm) above the near-field probe at the SiC/gold interface. Right inset: PSHetnear-field S3 image (width 1 µm) of the interface with sample and reference nanoFTIR locations indicated. b) Lightning rodmodel predictions for near-field signal from SiC using optical constants measured by in-house ellipsometry. While insensitive todetails of probe geometry (see text), fully retarded (Ret.) calculations provide superior agreement to the experimental spectrathan the quasi-static (QS) approximation.

VI. THE STRONGLY RESONANT LIMIT:SILICON CARBIDE

We can critically evaluate the generality of the light-ning rod model formalism through comparison with mea-surements of crystalline SiC, a strongly resonant mate-rial in the mid-infrared owing to an exceedingly strongsurface optical phonon at ω ≈ 950 cm−1. Here we findthe limit at which contingent assumptions for alternativenear-field models9,26,32,34,35 are expected to break down,since resonant materials can interact non-perturbativelyalong the entire length of the near-field probe. Thisbreakdown signals the importance of both probe geom-etry and field retardation effects. Lacking these con-siderations, previous models have dramatically overesti-mated the near-field contrast generated by SiC17,58, leav-ing the estimation of optical properties through quanti-tative analysis of near-field observables quite ambiguous.

Fig. 6 displays quantitative agreement between newlypresented nanoFTIR spectroscopy of a 6H SiC crystaland lightning rod model predictions. Asymmetry in theobserved phonon-induced probe-sample resonance spec-trum mimics that of the underlying surface responsefunction β(ω). To ensure unambiguous comparison be-tween experiment and theory, uniaxial optical constantsof our crystal were directly determined by in-house in-frared ellipsometry and were found consistent with lit-erature data for similar crystals59. A 100 nm gold filmwas deposited onto the crystal surface to provide a nor-malization material for nanoFTIR measurements, whichwere conducted at 60 nm tapping amplitude across theSiC-gold interface. The right inset of Fig. 6a displaysstrong interfacial contrast in near-field amplitude mea-

sured across the interface by pseudoheterodyne (PSHet)imaging11 with a CO2 laser tuned to 890 cm−1, withnanoFTIR acquisition positions indicated. As confirmedby nanoFTIR, near-field resonance with the SiC sur-face optical phonon produces a stronger signal than goldacross a considerable energy range, 800-940 cm−1. Suchstrong near-field resonances enable potential technolog-ical applications for guiding and switching of confinedinfrared light within nanostructured polar crystals, assuggested in related work60.

Predicted spectra presented in Fig. 6b reveal that ex-plicit consideration of field retardation effects accordingto the findings of Sec. IV (spectra labeled Ret.) sig-nificantly improves quantitative agreement with exper-imental spectra in contrast to the quasi-static predic-tion (labeled QS ), which drastically overestimates thenear-field contrast of SiC up to a factor of 20 over gold.The QS curve additionally reflects an excessive red-shiftof the probe-sample resonance peak on account of theoverly predominant low-momentum phonon excitationspermitted in the quasi-static approximation; these re-side at lower energy due to the typical positive groupvelocity of surface phonon polaritons. We furthermoreexplored the influence of particular probe geometries onthe predicted near-field spectrum by employing chargedistributions Λ(q, z) calculated for both the ideal hyper-boloidal probe geometry as well as for the actual profileof an used probe tip, obtained from an SEM micrograph(displayed as the blue curve in Fig. 1a). The Fig. 6bcomparison of SiC spectra predicted with these geome-tries reveals that only essential features of the probe ge-ometry, such as the overall conical shape and taper angle(θ ≈ 20) shared by both, are relevant for predicting

11

near-field contrasts at the 10% level of accuracy. Fur-ther quantitative refinements to near-field spectroscopywill therefore benefit from the standardization of repro-ducible probe geometries61.

VII. NANO-RESOLVED EXTRACTION OFOPTICAL CONSTANTS

Systematic improvements in the light sources anddetection methods available for near-field spectroscopynow enable sufficiently high signal-to-noise levelsand fast acquisition times for routine, reproduciblemeasurements7,8. Fig. 7 displays newly presentednanoFTIR measurements acquired on a 300 nm SiO2 filmwith silicon used for normalization, displaying both theamplitude S and phase φ of the probe’s back-scatteredradiation demodulated at the 2nd and 3rd harmonics ofthe probe frequency, collected at 60 nm tapping ampli-tude. Such broadband data are ideally eligible for thequantitative extraction of SiO2 optical constants in thevicinity of the transverse optical phonon (ωTO ≈ 1075cm−1).

Using the lightning rod model, a method requiring min-imal computational effort was developed to solve the in-verse problem of near-field spectroscopy, proceeding asfollows: The connection between optical properties of asample material (e.g. the complex dielectric function,ε = ε1 +iε2) and near-field observables (e.g. S and φ, orequivalently the real and imaginary parts of the complexback-scattered signal s = s1 + is2 at a given harmonicn ≥ 2) is described by a smooth map NF : C → C,with C the set of complex numbers. A “trajectory” s(ω)through the space of observable near-field signals there-fore corresponds to a trajectory ε(ω) through the spaceof possible optical constants. The uniqueness of this cor-respondence was confirmed for bulk and layered samplegeometries by computing s = NF(ε) across the parameterrange of interest for real materials (ε2 > 0) and ensuringlocal invertibility of the map, conditional on the deter-minant of the Jacobian matrix of NF:

|J(ε1, ε2)| > 0 with J(ε1, ε2) =∂(s1, s2)

∂(ε1, ε2). (23)

Because parameters internal to the operation of NFare frequently variable (e.g. sample thickness, tip radiusof the probe, tapping amplitude), instead of establish-ing the inverse map NF−1 as a “look-up table” by brutecomputation, we instead introduce a method for nucle-ated growth of the trajectory ε(ω) which optimizes con-sistency with the forward mapping s = NF(ε) beginningat some initial frequency ω0. We re-imagine the problemas a particle navigating ε-space under the influence of ex-ternal forces penalizing displacements δs from measuredsignal values s(ω). The trajectory ε(ω) for such a particlesolves, for example, the equation of motion for a damped

SiO2 Si

Esca Einc

300nm

FIG. 7. Amplitude S and phase φ of the back-scattered near-field signal from a 300 nm SiO2 film, measured in the vicinityof the surface optical phonon and referenced to the siliconsubstrate, as obtained in a single acquisition by nanoFTIR.

harmonic oscillator equilibrating to s = NF(ε):

d2

dω2δs+ 2ζ Ω

d

dωδs+ Ω2 δs = 0 (24)

with δs(ω) ≡ s(ω)−NF(ε(ω)

).

Here ζ denotes a damping constant tuned to induce crit-ical damping (ζ=1), and Ω is a force constant ensuringdecay to equilibrium over an interval δω = 2π/Ω compa-rable to the frequency resolution of measurement. Thisequation of motion enables adiabatic tracking of exper-imentally observed signal values while both penalizingdeviations δs and dissipating their energy. Eq. (24)may alternatively be parametrized by an auxiliary in-dependent variable x for which ω(x) increments onlywhen |δs(ω)| < δsthresh, a threshold value ensuring sys-tem equilibration arbitrarily close to the measured signalvalue at each ω. This also ensures solutions to Eq. (24)are relatively insensitive to the “guessed” initial condi-tion ε(ω0), amounting to a robust relaxation method.

Our inversion of measured s(ω) consists of numericallysolving Eq. (24) for ε by finite difference techniques62.This requires at least five evaluations of NF per ω- orx-step in order to estimate local first and second deriva-tives of NF with respect to real and imaginary parts ofε. Although consequently the procedure is more com-putationally costly than forward evaluation by the light-ning rod model, it is at least as efficient in principle asglobal nonlinear least-squares methods (e.g. Levenberg-Marquardt63) and often considerably faster, furthermorerequiring no a priori knowledge for the form of the fit-ting function. This is considerably advantageous in caseswhere spectra are not available in a sufficiently wide fre-quency range to permit well-determined fitting to ε(ω)by Kramers-Kronig-consistent oscillators64.

We applied our inversion technique to the spectro-

12

a) b) a) b)

FIG. 8. a) Typical variation in optical constants among thermal oxide thin films taken from literature ellipsometry. Pairs ofred and blue curves with identical line style are associated with distinct thin film samples.42 b) Optical constants of a 300 nmSiO2 film extracted from near-field spectra S2(ω) and S3(ω) following the method of Eq. (24).

scopic data displayed in Fig. 7 by parametrizing NFwith the reflection coefficient of an “unknown” 300 nmlayer (film thickness determined by AFM) on siliconsubstrate. For mapping the film’s optical constantsεfilm(ω) = ε1(ω) + i ε2(ω) to a measurable, normalizednear-field spectrum sn(ω), the form for NF used here isthat given by the lightning rod model, namely:

NF(ε1(ω), ε2(ω)

)= s film

n (ω)/sSin (25)

with

s filmn (ω) =

∫ π/Ω

−π/Ωdt sin (nΩt)E film

rad (d, ω), (26)

E filmrad (d, ω) = ~erad ·G film(d, ω)

~Λ0

I−ΛG film(d, ω), (27)

and d = A (1 + sin (Ωt)). Describing the near-field re-sponse of the film, G film(d, ω) is given by Eq. (11) interms of the film reflection coefficient r film

p (q, ω), whichis in turn a function of εfilm(ω) via Eq. (21). The sil-icon normalization signal sSi

n is computed analogously,but using the reflection coefficient for a bulk surface withfrequency-independent dielectric constant εSi ≈ 11.7. Allother parameters are defined as detailed in Sec. II.

In Fig. 8 we present the favorable comparison ofour extracted εfilm(ω) with typical literature optical con-stants for three thermal oxide films measured by conven-tional infrared ellipsometry42. Fig. 8a makes clear thetypical variation in optical constants expected among ox-ide films grown even under nominally fixed conditions.Furthermore, our extraction technique produced virtu-ally identical output when conducted on both 2nd and3rd harmonic near-field spectra (s2(ω) and s3(ω)), at-testing to the internal consistency of the lightning rodmodel.

Although near-field inversion has been very recentlydemonstrated on measurements of prepared polymers,

the existing technique relies on a polynomial expansionin β strictly limited to weakly resonant samples, vis. theperturbative limit of Eq. (13), and employs a modelwith tunable ad hoc parameters30,32. Our procedure re-moves both shortcomings. These advantages make Eq.(24) a suitable technique for the unconditional on-lineanalysis of near-field spectroscopy data in a diagnos-tic setting. Combining for the first time the powerfulnanoFTIR instrumentation with a quantitative inver-sion methodology unlimited by sample characteristics,this procedure makes possible potent new applicationsof nano-spectroscopy to the quantitative optical studyof phase-separated materials17,18 and nano-engineereddevices21,22, as well for the nano-resolved chemical iden-tification of structures in biological or surface scienceapplications7,9,16,32.

VIII. CONCLUSIONS AND OUTLOOK

The lightning rod model provides a general quantitativeformalism for predicting and interpreting the experimen-tal observables of near-field spectroscopy. Simplified de-scriptions of the probe-sample near-field interaction suchas the point dipole model can be obtained as special casesresulting from convenient though unnecessary physicalassumptions. In particular, the choice of effective probelength L30,32 was shown to be ad hoc in the quasi-staticapproximation, and consequently susceptible to dubiousa posteriori fitting to experimental data.

We find a fully electrodynamic treatment renders theeffective length construct unnecessary, since field retarda-tion effects modify the distribution of probe charge inter-acting with the sample. While this provides a resolutionto problems of convergence inherent to the quasi-statictreatment, sample-induced dielectric loading of strongantenna resonances (e.g. for the long ellipsoidal probe)was found to deceivingly modulate relative material con-

13

trasts predicted in the vicinity of sample resonances,such as the surface optical phonon of SiO2, an impor-tant caveat and consideration for the rational design ofoptimized spectroscopic probes47. Nevertheless, fine de-tails of the probe geometry for realistic conical probegeometries are predicted to impact observable near-fieldmaterial contrasts at or below a 10% level of variation.

Using the fully retarded lightning rod model with a real-istic probe geometry, we obtain quantitatively predictiveagreement compared both with tunable QCL near-fieldspectroscopy of SiO2 films with varying thickness andwith newly presented nanoFTIR spectroscopy measure-ments of the strongly resonant polar material SiC. Thisexhibits our model’s proper momentum-space descriptionof the probe-sample optical interaction, as well as its suit-ability for the truly quantitative description of stronglyresonant near-field interactions, in contrast with the ca-pabilities or implementations of the alternative modelsheretofore demonstrated.34,35,58

Finally, we present a deterministic method to invertthe lightning rod model without recourse to ad hoc pa-rameters or over-simplifications. This rather generaltechnique flexibly solves the inverse problem of near-field spectroscopy at a computational cost significantlylower than exhaustive lookup-tables or oscillator fit-ting methods, offering exciting opportunities for the on-line interpretation of nano-resolved near-field spectra ac-quired in a diagnostic setting. We envision the in-verse lightning rod model employed quantitatively fordeeply sub-wavelength optical studies of naturally orartificially heterogeneous and phase-separated materi-als, promising further novel applications to systems likeenergy storage nanostructures,65 transition metal oxideheterostructures,66 and single- or multi-layered grapheneplasmonic devices20,21.

There remain outstanding challenges for the presentmodel, including its extension to cases where deviationsfrom axisymmetry are crucial, as for s-polarization ofincident light, or for probe geometries with strong ro-tational asymmetry. We envision an expansion of ourboundary element methods and of the lightning rodmodel into basis components with differing rotational“quantum numbers”67 to capture the features of irrota-tional geometries in a computationally inexpensive fash-ion. Furthermore, the explicit application of our modelto dielectric probes, particularly in the plasmonic regime,is an undertaking of great potential interest for which theextension of our electrodynamic boundary element (Ap-pendix D) to materials of non-negligible skin depth mightplay a crucial role. However, even at its present stagethe quantitative scattering formalism presented here alsolays a solid foundation for the rational analysis and opti-mization of tip-enhanced optical phenomena in an ever-growing number of exciting experimental applications,including single-molecule Raman spectroscopy15,75 andother novel partnerings of optics with scanning tunnel-ing microscopy.76,77

ACKNOWLEDGMENTS

This work was supported through NASA grantsNNX08AI15G and NNX11AF24G. Alexander S. McLeodacknowledges support from a U. S. Department of EnergyOffice of Science Graduate Fellowship.

Appendix A: Experimental methods

In the following sections, we apply the lightning rodmodel in comparison with near-field spectra measuredfor SiO2 thin films and SiC, acquired with the follow-ing experimental apparatus. Infrared nano-imaging andnano-spectroscopy measurements were performed with aNeaSNOM scanning near-field optical microscope (Nea-spec GmBH ) by scanning a platinum silicide AFM probe(PtSi-NCH, NanoAndMore USA; cantilever resonancefrequency 300 kHz, nominal radius of curvature 20-30nanometers) in tapping mode over the sample while il-luminating with a focused infrared laser beam, result-ing in back-scattered radiation modulated at the probetapping frequency Ω and harmonics thereof. In ourpseudo-heterodyne detection setup, this back-scatteredradiation interferes at a mercury-cadmium-telluride de-tector (Kolmar Technologies Inc.) with a reference beamwhose phase is continuously modulated by reflection froma mirror piezoelectrically oscillated at a frequency δΩ(≈ 300 Hz). Demodulation of the detector signal atfrequency side-bands nΩ ±mδΩ for integral m suppliesthe background-free amplitude Sn and phase φn of theinfrared signal at harmonics n of the probe’s tappingfrequency.4,10,11

The super-linear dependence of near-field interactionsversus the tip-sample separation distance implies that, inthe case of harmonic tapping motion, signal harmonics atn ≥ 2 are directly attributable to near-field polarizationof the tip48. Contrasts in near-field signal intensity andphase at these harmonics therefore correspond to vari-ations in local optical properties of the sample26. Tun-able fixed-frequency CW quantum cascade lasers (QCLs,Daylight Solutions Inc.) and a tunable CO2 laser (Ac-cess Laser Co.) were used for imaging and spectroscopyof SiO2 films and SiC, respectively.

NanoFTIR spectroscopy78 was enabled by illuminationfrom a broadband mid-infrared laser producing tunableradiance across the frequency range 700-2400 cm−1. Thiscoherent mid-infrared illumination is generated throughthe nonlinear difference-frequency combination of beamsfrom two near-infrared erbium-doped fiber lasers – oneat 5400 cm−1 and the other a tunable supercontinuumnear-infrared laser (TOPTICA Photonics Inc.) – result-ing in 100 fs pulses at a repetition rate of 40 MHz. Anasymmetric Michelson interferometer with 1.5 millimetertravel range translating mirror enables collection of de-modulated near-field amplitude Sn(ω) and phase φn(ω)spectra with 3 cm−1 resolution.

14

Appendix B: Resolution of the field from a chargedring into evanescent waves

The xy-plane Fourier decomposition of the Coulombfield of a point charge Q located at the origin is wellknown68:

~E(~r) = − Q2π

∫∫ ∞−∞

dkx dky

(ikxx+ ky y

q+ z

)×ei(kxx+kyy)+qz

= −Q∫ ∞

0

dq q (J0(qρ)z + J1(qρ)ρ) eqz (B1)

for z < 0 and with q ≡√k2x + k2

y. This decomposition

can be applied similarly to a ring of charge with radiusR, centered in a plane through the origin with z-axisnormal:

~ER(~r) =Q

4π2

∫ 2π

0

dφ′∫ ∞

0

dq

∫ 2π

0

dφ ~ER(q, ~r, φ′)

~ER ≡ −(z + i

kxx+ ky y

q

)eiq(ρ cosφ−R cos (φ−φ′))+qz.

Here φ′ is an angular integration variable about the cir-cumference of the ring. We obtain

~ER(~r) = −Q∫ ∞

0

dq q J0(qR) (J0(qρ)z + J1(qρ)ρ) eqz.

(B2)The total field is thus a sum of axisymmetric p-

polarized evanescent waves weighted by the geometry-induced prefactor q J0(qR). Eq. (B2) constitutes thecentral result of this section.

Appendix C: Quasi-static boundary element methodfor an axisymmetric dielectric and conductor

A tractable electrostatic boundary element method ap-plicable to systems of homogeneous dielectrics can be de-veloped as follows67. Gauss’s law constrains the densityof bound charge ρb at the boundaries between dielectricmedia as:

∇ · ~E = ∇ ·~D

ε= 4πρb

∴ 4πρb = δS

(1

ε2− 1

ε1

)n12 · ~D (C1)

Eq. C1 follows in the case that free charge is absent at

dielectric boundaries such that ∇· ~D = 0, and δS is a sur-face Dirac delta function associated with the boundarybetween media of dielectric constant ε1 and ε2, with n12

the unit vector perpendicular to this boundary orientedfrom medium 1 to medium 2. Continuity of the surfacenormal displacement field across the dielectric interfacepermits its evaluation at positions ~r along the boundaryas a limit taken infinitesimally inside medium 2:

n12 · ~D (~r) = ε2 limt→0+

−n12 · ∇V (~r + t n12) . (C2)

The scalar potential V (~r) finds contributions from both

incident (external) fields ~Einc, originating as from dis-tant free charges, as well as from bound charges at thedielectric boundary. Taking the bound charge ρb as theproduct of a surface density σQ (distinguished from elec-trical conductivity σ) with the surface Dirac delta func-tion, the latter contribution comprises a surface integralon the boundary S:

Vb(~r) =

∫S

dS′σQ(~r′)|~r − ~r′| (C3)

Evaluating the discontinuous surface normal electric field−n12 · ∇Vb from this contribution involves:

limt→0+

−n12 · ∇ (1/ |~r + t n12 − ~r′|) = 2πδ(~r − ~r′)− F (~r, ~r′)

with F (~r, ~r′) ≡ − n12 · (~r − ~r′)|~r − ~r′|3

. (C4)

Gauss’s law (Eq. C1) therefore yields an integral equa-tion in the surface bound charge density σQ(~r):

4πε1ε2ε1 − ε2

σQ(~r) = ε2

[n12 · ~Einc(~r) + 2πσQ(~r)

−∫S

dS′ F (~r, ~r′)σQ(~r′)], (C5)

which upon consolidation yields:

2πε1 + ε2ε1 − ε2

σQ(~r) = n12 · ~Einc −∫S

dS′ F (~r, ~r′)σQ(~r′).(C6)

Without loss of generality, this equation can be utilizedto pre-compute the quasi-electrostatic response of an ax-isymmetric body of dielectric constant ε2 to incidentfields, taking ε1 = 1 as air, parametrizing the integral ker-nel F by axial and surface radial coordinates z and Rz,respectively, and expressing n12 through axial derivativesof the latter.

However, to unambiguously present our method of so-lution to equations like Eq. C6 and to promote its ap-plication for the description of metallic near-field probes,we confine our attention specifically to the ideally con-ducting limit, wherein ε2 is divergent. For Eq. C1 tohold with finite normal displacement in Eq. C2 there-fore requires a vanishing normal gradient of the total po-tential V just inside the probe surface. Lacking free orbound charges within its volume, the probe interior andsurface therefore reside at constant total potential, sig-nifying zero internal field and perfect screening by thesurface:

Vinc(~r) + Vb(~r) = V0 on S. (C7)

This criterion follows equivalently from Eq. (C6) in thelimit ε2 ε1 through reverse application of Eq. (C4).

The incident potential of an axisymmetric evanescentfield is given in cylindrical coordinates ρ, φ, and z by

15

Vinc(~r) = J0(qρ)/q e−qz. The potential Vb is gener-ated by the surface charge density σQ(~r), which maybe divided into a continuum of rings, each with chargedQ = λQ(z) dz and radius Rz:

Vb(~r) =

∫S

dS′σQ(~r′)|~r′ − ~r|

=

∫ ∞0

dz′Φ(~r, z′)λQ(z′) (C8)

Φ ≡∫ 2π

0

dφ′

2π

1√(z′ − z)2 + ρ2 +R2

z′ − 2ρRz′ cosφ′

=2K(− 4 ρRz′

(ρ−Rz′ )2+(z−z′)2 )

π√

(ρ−Rz′)2 + (z − z′)2. (C9)

Here Φ constitutes the Coulomb kernel for a ring ofcharge, and K(. . .) denotes the elliptic integral of thefirst kind. Evaluating Vobj at the boundary of the object(ρ = Rz) and discretizing z in Eqs. (C7) and (C8) as byGauss-Legendre quadrature, we obtain the linear system

M~λQ = V0 − ~Vinc (C10)

with Mij ≡ Φ(zi, zj) δzj .

Vectors denote evaluation at positions zi,R(zi). Thecondition of overall charge neutrality fixes the value ofV0: ∑

i

~λQ δzi = 0 =∑i

δzi M−1(V0 − ~Vinc

)

∴ V0 =

∑i δzi

[M−1 ~Vinc

]i∑

i δzi

[M−1 ~I

]i

. (C11)

Here ~I denotes a vector with all entries unity. While Eq.(C10) would appear to be directly solvable, such Fred-holm integral equations of the first kind are notoriouslyill-conditioned. Consequently, we adopt regularizationmethods6970 to invert the integral operator (matrix) M,yielding smooth functions λ(z) in accord with standardquasi-static solutions for well-studied geometries like theconducting sphere and ellipsoid. (It is worth noting that,since Eq. (C6) presents a well-conditioned Fredholm in-tegral equation of the second kind, no such regularizationof the solution is required in the case of a dielectric solid.)Once the inverse operator M−1 has been computed for agiven geometry, calculation of λ(z) for arbitrary Vinc(~r)is fast and trivial.

For an axisymmetric system, Eqs. (C10) and (C11)together with this solution method are sufficient to cal-culate the linear charge density induced on a conductingbody due to an incident quasi-static field, and constitutethe central result of this section. In practice, the con-verged calculation of M−1 for a particular axisymmetricgeometry takes no longer than a few tens of seconds on asingle 2.7 GHz processor. Calculation of λ(z) for a rangeof q values sufficient for converged lightning rod modelcalculations requires only several seconds using the sameprocessor.

Appendix D: Electrodynamic boundary elementmethod for an axisymmetric conductor

As in the quasi-static case, the charge distribution in-duced on a nearly perfectly conducting object by an in-cident electrodynamic field oscillating at frequency ω re-sides exclusively at the object’s surface. To compute thisdistribution, we begin with detailed force balance at theboundary S along directions tangential to the surface.Assuming axisymmetry, we need only consider without

loss of generality the surface tangential directions ξ or-

thogonal to φ that possess positive z-component:

ξ ·(~Einc + ~Eobj

)= ~0 on S. (D1)

Since ~E = −∇V + iω ~A for scalar and vector potentials

V and ~A, we have

Einc ξ(~r) =

∫S

(∂ξ dVobj(~r)− iω ξ · d ~Aobj(~r)

)on S

=

∫ L

0

dz′[∂ξΦ(z, z′)λQ(z′)

−iωAξ(z, z′) I(z′)], (D2)

where we have parametrized points on S by the object’saxial coordinate 0 < z < L; meanwhile λQ(z) ≡ dQ/dzdenotes the linear charge density and I(z) denotes thetotal current passing along the object surface through az-normal plane at z. Φ and Aξ denote integration kernelsfor the scalar and vector potentials, respectively.

The continuity equation for charge implies ∂zI(z) =iω λQ(z), and since current is forbidden to flow fromthe hypothetically isolated object, integration by partsyields:

Einc ξ(z) =

∫ L

0

dz′λQ(z′)[∂ξΦ(z, z′)

−ω2

∫ z′

0

dz′Aξ(z, z′)]

(D3)

In terms of the azimuthal angle φ and surface radial co-ordinate at z denoted Rz, the scalar potential kernel isgiven by

Φ(z, z′) =

∫ 2π

0

dφ′

2π

eiω/c∆(z,z′,φ)

∆(z, z′, φ)(D4)

with ∆ ≡√

(z − z′)2 +R2z +R2

z′ − 2RzRz′ cosφ,

which may be computed straightforwardly for a given ob-ject geometry by adaptive quadrature. The exponentialphase ensures the integrand is evaluated at retarded time.The vector potential kernel may be established from

~A(~r) =1

c2

∫dS′

~K(~r′)|~r − ~r′| e

iω/c |~r−~r′| (D5)

and Aξ(z, z′) ≡ ξz ·d ~A

dz′(z), (D6)

16

with ~K(z) ≡ I(z)/2πRz ξ denoting the local surface cur-

rent. Noting that dS′ = 2πR′z√

1 + ∂z′R′z dz′ and that

the direction of ξ is manifestly z- and φ-dependent (ex-

pressed here as ξzφ), we obtain

dAξdz′

(z) =√

1 + ∂z′R′zI(z′)

c2

∫ 2π

0

dφ′

2π

ξzφ · ξz′φ′

|~r − ~r′|ei ...,

with the exponential factor unchanged. The φ-dependence of Aξ is rendered moot on account of ax-isymmetry, and so is suppressed.

Finally, since the surface tangential unit vector atheight z and azimuthal angle φ is expressed in terms ofthe radial coordinate Rz and the radial unit vector ρφ as

ξzφ =∂zRz ρφ + z√

1 + ∂zR2z

, with ρφ · ρφ′ = cos(φ− φ′),

we obtain the vector potential kernel as:

Aξ(z, z′) =1

c2

[∫ 2π

0

dφ′

2π

eiω/c∆(z,z′,φ′)

∆(z, z′, φ′)+

∂zRz ∂z′Rz′∫ 2π

0

dφ′

2π

eiω/c∆(z,z′,φ′)

∆(z, z′, φ′)cosφ′

]

× 1√1 + ∂zR2

z

. (D7)

Here ∆ is defined as in Eq. (D4), and note that the firstterm within brackets in fact equates with the scalar po-tential kernel. Only the second term must be computedanew, and similarly by adaptive quadrature.

We now define a convenient quasi-potential functionfor the incident field:

Vinc(z) ≡ −∫dz√

1 + ∂zR2z ξz · ~Einc(z)

= −∫dz (∂zRz Einc ρ + Einc z) . (D8)

Proceeding with Eq. (D3), we relabel z → z before ap-

plying the operation∫ ξz

0dξ =

∫ z0dz√

1 + ∂zR2z to both

sides, resulting in:∫ L

0

dz′λQ(z′)

[Φ(z, z′)− ω2

c2

∫ z

0

dz

∫ z′

0

dz′ Aξ(z, z′)]

= V0 − Vinc(z). (D9)

Here we have applied Eq. (D8) and taken V0 as a con-stant of integration. Furthermore we have defined a newvector potential kernel Aξ(z, z′) ≡

√1 + ∂zR2

z Aξ(z, z′),which is now symmetric in its two arguments. Notethat the first term in brackets in Eq. (D9) accountsfor the retarded Coulomb force among surface charges,whereas the second term describes radiative forces withstrength of order O2(L/λ) produced by conduction cur-rents, where λ the free-space wavelength of light.

As in Appendix C, discretizing z results in a linearsystem[

Φ− ω2

c2W

T AW

]W~λQ = V0 − ~Vinc, (D10)

where Φij ≡ Φ(zi, zj), W ≡ diagδzi, Aij ≡ Aξ(zi, zj),Wij ≡ δzi θ(j− i), and θ(. . .) denotes the Heaviside unitstep function. The superscript T denotes matrix trans-pose. Vectors again denote evaluation at axial and radialcoordinates zi,R(zi) along the object surface. Self-consistency requires a value of V0 ensuring charge neu-trality. Taking M to be the full integral operator preced-

ing ~λQ in the linear system above, V0 is again given by

Eq. (C11), and ~λQ is obtained via inversion of M. Notethat the particular selections of lower integration boundson Vinc in Eq. (D8) and A in Eq. (D9) are naturallyrendered arbitrary when this condition is satisfied. Asin the quasi-static case, once M−1 has been computedfor a given geometry (less than one minute of computa-tion on a 2.7 GHz processor), the calculation of λQ(z)

for arbitrary ~Einc(~r) is both fast and trivial (several mil-liseconds). To emulate plane wave illumination from aninclination angle θ with respect to the z-axis, in this workwe substitute the axisymmetric analog

~Einc(~r) =(J0(qρ) z + i

√k2 − q2

qJ1(qρ) ρ

)× e−i

√k2−q2z (D11)

with q ≡ k sin θ and k ≡ ω/c.

This field profile equates with a rotational sum of θ-directed plane waves inbound from all azimuthal anglesφ.

For an axisymmetric system, Eqs. (D4), (D7), and(D10) are sufficient to calculate the linear charge densityinduced on a conducting body due to an incident elec-trodynamic field, and constitute the central result of thissection. In practice, the converged electrodynamic cal-culation of M−1 for a particular axisymmetric geometrytakes only twice as long as for the quasi-static case.

Appendix E: Radiation from an axisymmetricconductor

The far-field radiation profile from an arbitrary cur-rent distribution can be obtained by integrating the far-

field contribution←→G FF to the Green’s dyadic function←→

G 71 from infinitesimal current elements at positions ~r′,here for demonstration considered oriented along the z-direction, as

d ~Erad(~r) =iω

4π~GFF,z(~r, ~r′) jz(~r′) dV ′ (E1)

with ~GFF,z(~r, ~r′) ≡ −1

c2eiω/c |~r−~r′|

|~r − ~r′| sin θ θ,

17

FIG. 9. (Left) The field radiated from an axisymmetric bodyto an observation point at inclination angle θ is constructedfrom the contributions of currents (shown in red) through in-finitesimal surface annuli. (Right) The two independent po-larizations composing any annular axisymmetric current dis-tribution, with associated angular radiation profiles (Eq. E1)shown schematically in blue.

exhibiting the familiar field profile of a radiating dipole.Here θ denotes the inclination angle of the observationpoint ~r from the z-axis in a spherical coordinate system.

The dimension of a nearly perfect conductor is by def-inition much greater than the magnetic skin depth of theconstituent material. Consequently, volume integrationreduces to an integral over surface current contributions

dS ~K(~r). In an axisymmetric object, these contributionsare associated with surface annuli located at axial coor-dinates z and radii Rz, for which dS = 2πRz dξ withdξ ≡

√1 + ∂zR2

z dz. We first evaluate the radiated fieldfrom such an annulus, considering contributions from thetwo independently allowed polarizations of axisymmetriccurrent separately, as depicted in Fig. 9.

We define the total current as Iα(z) = 2πRzKα(z) forpolarizations α = z, ρ. For the z-polarized contribution,integrating Eq. E1 through an annulus about azimuthalangle φ obtains

d ~Erad,z = − iω

4πc2eiω/c δr(z)

δr(z)sin θKz(z) dξ

×∫ 2π

0

dφRz exp (iω/cRz cosφ sin θ) θ

= − iω

4πc2eiω/c δr(z)

δr(z)sin θ Iz(z) dξ

× J0(ω/cRz sin θ) θ. (E2)

Here δr(z) denotes the distance from the center of thez-located annulus to the observation point, and we haveapplied the approximation |~r − ~r′|−1 ≈ δr(z)−1 valid forδr(z) Rz. An elementary analysis accounting for ro-tation of the radiant polarization vector in the integrand

of the ρ-polarized contribution similarly results in:

d ~Erad,ρ =ω

4πc2eiω/c δr(z)

δr(z)cos θ Iρ(z) dξ

× J1(ω/cRz sin θ) θ. (E3)

Current I(z) flows on the surface of an axisymmetricconductor along a surface tangent vector

ξ =∂zRz ρ+ z√

1 + ∂zR2z

.

Consequently, the total radiation from the axisymmetricbody of length L is given by a commensurate sum of z-and ρ-polarized contributions:

~Erad(θ) = − iω

4πc2eiω/c∆r

∆r

∫ L

0

dz E(z, θ) I(z) θ (E4)

with E = e−iω/c z cos θ[

sin θ J0(ω/cRz sin θ)

+ i ∂zRz cos θ J1(ω/cRz sin θ)]. (E5)

Note the factor 1/√

1 + ∂zR2z has been absorbed by the

integration measure dz. Here ∆r ≡ δr(z) + z cos θ isthe distance from one apex of the object (at z = 0)to the observation point, and we have applied the ap-proximation δr(z)−1 ≈ ∆r−1 appropriate for distances∆r L. After applying the continuity equation forcharge ∂zI(z) = iω λQ(z) (with λQ the linear chargedensity) together with the fact that the current van-ishes at the extrema of a hypothetically isolated body(at z = 0, L), integration by parts yields:

~Erad(θ) = − ω2

4πc2eiω/c∆r

∆r

∫ L

0

dz λQ(z)

∫ z

0

dz′E(z′, θ) θ.

(E6)Provided an electrodynamically consistent charge dis-

tribution λQ(z) calculated at frequency ω, Eq. (E6) canbe evaluated straightforwardly for a given object geom-etry by quadrature. In the notation of Appendix D, the

complex amplitude of the θ-polarized radiation field be-comes:

Erad(θ) = − ω2

4πc2eiω/c∆r

∆r~λTQ W W ~Eθ. (E7)

Here the superscript T denotes vector transpose, and ~Eθindicates evaluation of E(z, θ) at the chosen observationangle θ. Together with Eq. (E5), this expression is suf-ficient to compute the electric field radiated from a con-ducting axisymmetric system, and constitutes the centralresult of this section.

Projected onto a detector sensitive to θ-polarized light,the radiation contributions [~erad]i utilized in Sec. II arecomputed by expressing each single-momentum probe re-sponse function (linear charge density) Λ(qi, z) in dis-

cretized real space representation[~λQ]j≡ Λ(qi, zj) and

applying Eq. (E7), taking θ ≈ 60 relative to the z-axisof the near-field probe as the typical collection angle ofexperimental detection optics.

18

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