Near-Field Optics of Localized Surface Plasmons

viHowever, one important part of the instrument, namely the tip-sample distance control mecha-nism based on shearforce, spans a whole chapter, i.e. Chapter 4. There a thorough analysis ofthe shearforce sensor, the quartz crystal tuning fork, and the estimation of some characteristicquantities expressing the actual instrument performance, have been worked out. In particular,the eect of the quality factor of the tuning fork oscillator on the scan speed, and the possibilitiesto change it by both mechanical and electrical means, have been investigated and reported. Thiswill hopefully help the future users to improve the current PSTM setup.With Chapter 5 the presentation of the experimental results begins, starting with the opticalNF characterization of the elementary unit of our samples, namely a single oblate gold particle.The optical signature of this unit has been identied and discussed, in the two dierent linearpolarization cases, TM and TE, and successfully compared with GDT simulations. As the ideabehind the presentwork is to provide the basic knowledge for the development of devices for lightmanipulation and transport, a spectroscopic study of the single particle response across the red-very near infrared spectral region has also been performed. This showed an increase of the eectiveNF intensity for increasing wavelength in that region.The single particles are aimed to constitute the bricks of the optical nanodevices mentioned atthe beginning. In these systems an appropriate ensemble of similar such particles will be built up,in which the interactions between the dierent units will play a fundamental role. For this reasonit is of major importance to investigate the eects of interacting metal nanoparticles. This wasdone in the following chapters, where 1D and 2D arrangements of gold nanoparticles (chainsand gratings, respectively) were studied. With the chains (Chapter 6 ), the eects of both NFand FF coupling were probed. The former turned out to be of quite dicult interpretation in thePSTM images, due to the high sensitivity of the NF particle coupling to even small deviations ofthe chain structure from a perfectly uniform linear array (as also conrmed byGDTsimulations).The latter was correctly interpreted in the frame of diraction theory, a picture which showedclearly to t with the experimental results.Therefore, when wewent on with the 2D gratings (Chapter 7)we focused on the simple case oflarge (above half the wavelength) particle spacing, i.e. FF particle coupling only. The complicationdue the higher degree of freedom in system geometry was rst approached byinvestigating a gratingof limited size, namely 33 particles. For this system we could observethedevelopment of particleinteraction towards the formation of an eective extended grating, characterized by a collectivebehavior of the particles. While both qualitative and quantitative analysis was in agreementwiththe GDT simulations, still it was found that, due to the peculiar NF distribution, the observedNF intensity exhibits a crucial dependence on several geometrical parameters (particle spacing,eective observation height) which determine the actually measured value. Indeed, with PSTMinvestigations of extended 2D gratings the consequences of the FF grating eects on the NFpattern could be found, whichshowed not to be much prominent. Complications in interpretationof the images arise from the inclination of the exciting light beam required for PSTM, comparedto the measurements performed by extinction spectroscopy at normal incidence, whose eect wetried to point out and make clear.Finally, in Chapter 8 another example of combined application of NF optical microscopy andplasmon coupling in metal nanostructures is presented, which also ts to the idea of the fabricationof optical nanodevices based on these materials. Namely, the propagation of a surface plasmonlaterally conned to a nanowire of sub-wavelength size is demonstrated. This could allow thesolution of the next task on the way to the fabrication of such devices, i.e. interfacing the lightsignal processing units, based on ensembles of noble metal particles, with the outer world ofconventional FF optics, by guiding the optical signal input and output to and from these unitswith similar sub-wavelength sized metal waveguides.
Chapter 1Neareld optics
1.1 IntroductionThe capability to image samples with very high magnication is a key demand in many scienticdisciplines, e.g. in life- as well as in materials science, both for fundamental research and forpractical applications. Traditionally, due to historical reasons and ease of operation the opticalmicroscopes have been the most widely used instruments, despite the limitations in resolution (seeSection 1.2). During the last century, rst the electron based microscopes, namely TransmissionElectron Microscope (TEM) [14] and Scanning Electron Microscope (SEM) [15], have increasedthe attainable resolution in the 1930s. This was possible thanks to the particlewave dualism ofelectromagnetic radiations: while lightismade of photons, the electrons can be seen in turn aswaves with de Broglie wavelength  = h=mv (h Planck constant, m and v mass and speed of theelectrons, respectively). From this relation we can derive that if the particles used as a probe for thesample are electrons instead of photons, for the same single particle energy the involved wavelengthis much smaller: for example at 1 eV a photon has a wavelength 
=1.2 m while for an electrone=1.2 nm. Therefore, whereas the diraction limit to the resolution still holds its numericalvalue is in practice much smaller. Later on, in the 1980s the development of new scanning probetechniques, namely Scanning Tunneling Microscope (STM) [16] and Atomic Force Microscope(AFM) [17], has allowed to reacheven atomic resolution, mainly based just on the probe tip size.On the other hand, all the above mentioned non-optical microscopes either are invasive or putstrict constraints on the sample properties or preparation techniques. Moreover, the possibilitytoperform investigations of optical spectroscopy and to study optical polarization properties havebeen lost in these instruments. For this reason, in the last 20 years a growing interest has beendevoted to the so-called neareld optical microscopy, originated from a gedanken experimentbySinge [18] put into practice only almost 60 years later, by means of the rst, original ScanningNeareld Optical Microscope (SNOM) setup [19,20]. Neareld (NF) optical microscopy triesto put together the local probe principle and the scanning technique of STM and AFM with thecapabilities associated with a light probe. More and more variants have been appearing over theyears, and nowadays we really begin to speak of NF optical spectroscopy(seee.g.Refs. [21,22]).
1
2 1 NEARFIELD OPTICS
1.2 Limits of conventional optical microscopyLet us consider an optical microscope which is used to image a pinhole of diameter a illuminated intransmission by a plane waveofwavelength . When a  the light from the pinhole is collectedat a low diraction angle . When a decreases  increases, up to lling the whole half-space ofthe detector for a = . For a<the angle is not allowed to increase further and the microscopewill not be able to dierentiate the hole size, just stating it to be smaller than  /2. This isbecause we are dealing with a wavevector component parallel to the screen !k k whose modulusis increased to a value larger than 2= in air. Therefore, the corresponding waveisevanescent,i.e. no solution for propagation of the wave is possible. Hence, no more information is reachingthe microscope detector, as it stays conned at the sample object NF and does not propagate faraway. This strongly inuences the resolution limit of fareld (FF) optical microscopy, as it isshown in the following detailed mathematical discussion.Foramonochromatic light plane wave exciting a sample whose optical NF has to be imaged, theamplitude of the exciting electric eld is much greater than that of the exciting magnetic eld,in the well known ratio !E  = !B   c, with c speed of light. Moreover, when the sample is notferromagnetic as it is the case for e.g. gold structures (i.e. magnetic permeability sample=0 asfor vacuum and/or air) this low magnetic eld does not give rise to any signicant eect in thesample. In fact, even if electric currents of the gold free electrons could in principle arise due tothe coils that can be ideally traced inside the sample, its electromagnetic behavior in response tothe light excitation is largely dominated by the electric eld, i.e. by the electric polarization ofthe gold structures. Furthermore, on the side of detection, in NF optical microscopy performed incollection with a bare dielectric tip (as it is the case for our instrument, see Section 1.4) also theprobe is not sensitive to the magnetic eld of the sample. Indeed, glass is not magnetic (tip=0)and has no free charge carriers, such that no current can be induced inside it by any externalmagnetic eld. For the same reason the electric polarization of the tip is also limited comparedto that of a metallic sample, and the electric eld around the latter is in rst approximation thesame as if the tip was not there. This is the reason why results obtained with uncoated glass bertips can be reproduced by so-called non-global theories, only taking into account the sample eldand neglecting the tip, dierently from metal coated tips [23]. Therefore, the intensity I of thecollected light is generally associated to the square modulus of the electric eld:I / !E 2 (1.1)On the other hand, as a general result the electric eld of a wave !E (!r ) at a position !r k =(x; y)in a plane z = const: such that !r =(!r k;z) can be expressed by means of a Fourier integral as:!E (!r )= 12 Z f!E (!k k;z) ei!k k
!r k d!k k (1.2)where k =2=0n is the modulus of the wavevector !k , with 0 wavelength of light in vacuumand n refractive index of the considered medium, and !k k =(kx;ky). f!E (!k k;z) is called
1.2 LIMITS OF CONVENTIONAL OPTICAL MICROSCOPY 3
two-dimensional Fourier spectrum of !E (!r ). The eld has to satisfy the following wave equation:(r2 + k2)!E (!r )=0 (1.3)that together with the boundary condition that the eld goes to zero far away:!E ! 0; z ! +1 (1.4)(i.e. the eld source is in the half-space z 0) allows to write the solution in the following form:!E (!r )= 12 Z !A (!k k) eikzz ei!k k
!r k d!k k (1.5)with kz = qk2  k2k and !A (!k k) being an arbitrary function. Therefore, the eld is expressed asa sum of plane waves with wavevectors !K =(!k k;kz), named angular spectrum of plane waves asthe quantities ki=k for i = x; y; z are the directional cosines of the plane waves wavevectors.From the above expression of kz one can distinguish two cases:kz = jkzj for kk < jkj ; (1.6)kz = i jkz j for kk > jkj (1.7)
leading in the former case to propagating (or homogeneous) waves in the direction of !k ,inthelatter case to evanescentwaves, exponentially decaying with distance from the plane z = const:,according to the factor ekzz.The presented discussion allows to derive the resolution limit inherenttoFFmicroscopy. Let ussuppose that we know the eld i.e. its Fourier spectrum at the plane of the sample, z =0. Thespectrum at an observation plane at z = s is then:f!E (!k k;d)=f!E (!k k; 0) eikzs (1.8)and the eld is: !E (!r k;s)= 12 Z !E (!k k; 0) eikzs ei!k k
!r k d!k k (1.9)but kz can be real or imaginary depending on kk, according to Eqs. 1.6,1.7. Concerning the planeof the sample, the correspondence between the size of the eld features in direct space and thesize in Fourier space of frequencies or wavevectors, is given by:kk = 2k (1.10)that is, the larger kk, the smaller the corresponding spatial dimension k. But we see that onlythe waves with wavevectors kk  k can propagate and reach a FF detector, while the informationsabout the smallest size features, corresponding to kk >kdo not propagate, but decay exponentiallyaway from the sample plane. This is the reason why NF microscopy puts the local probe that
4 1 NEARFIELD OPTICS
close to the sample, in order to provide superresolution. As the limit to the above mentionedcase is kk = k,itfollows that the resolution limit, i.e. the smallest spatial variation of the eldand so of the intensity that can be detected is
l  2 (1.11)Apart from some constants, characteristic of the instrumental implementation, this result is quali-tatively consistent with the notion of diraction limit approximately equal to half the wavelength,and with the respective formulations by either Abbe [24]orRayleigh [25]. The latter in particulargave it in the form known and used nowadays for optical microscopes:
lRayleigh =0:61 N:A: (1.12)where N:A: is the numerical aperture of the objective, N:A:= n sin , with n refractive index ofthe imaging medium, and  half angle of the light cone. The particular numerical coecient comesin this case from the criterion that the two objects to be resolved at the distance 
l will havethe intensitymaximum of one of the two coincident with the rst intensity minimum of the other.The value in Eq. 1.12 can be pushed down somehowby using immersion oil objectives with highern than air, suchthatN:A: can be made larger than 1, but we see that the approximate value of=2 holds.The way in which the diraction limit to optical FF microscopy resolution has been presented hereis that of Fourier optics. Alternative points of view for physical understanding of this phenomenonhave been proposed which are also instructive to see, as either the quantum theory way, derivingthe resolution limit from the uncertainty principle of Heisenberg for photons [26, 27], or the secondprinciple of Thermodynamics [26].1.3 Beyond fareld optical microscopy: the neareldIt was shown above that in conventional FF optical microscopy the resolution is diraction limited.When on the contrary we operate NF optical microscopy, this resolution limit can be overcome.Here the necessity of a denition of NF zone compared to the FF one comes out. Obviously, thedenition of an exact boundary between the two domains is always somehow arbitrary, as theelectromagnetic eld is just one entity. As a consequence, the eld changes continuously betweenthe two extreme NF and FF limits, which can be dened as r   and r  , respectively,wherer is the distance from the object of interest (i.e. sample or probe tip) and  is the wavelength ofthe involved (monochromatic) light beam. Both the needs to dene the NF zone and to providean example of dierentNFversus FF behavior can be conveniently satised when one considersthe electric eld of an oscillating electric dipole. This example also ts well with our samples ofpolarized metal nanoparticles, of which the isolated oscillating dipole can be considered a goodmodel in rst approximation (see Chapter 5). For an dipole which oscillates while keeping itsmomentum vector !p 0 along the direction of oscillation as:!p (!r;t)=!p 0 ei(!k 
!r !t) (1.13)
1.3 BEYOND FARFIELD OPTICAL MICROSCOPY: THE NEARFIELD 5The spatial dependence of the electric eld generated around such a dipole can be expressed inspherical coordinates as [25]: Er =2p0 cos r3 (1 ikr) eikr ; (1.14)E = p0 sin r3 (1 ikr  k2r2) eikr ; (1.15)E =0 (1.16)where  is the azimuthal angle (from the direction of oscillation) and r the distance from the dipolecenter. No dependence on the equatorial angle  shows up, i.e. the eld has cylindric symmetry.As both radial and transversal eld components Er and E exhibit a dependence on r of thetype  1=rn with n 1, it is clear that the eld strength changes much more rapidly in the NF(i.e. at small r, where jEij! +1;r! 0, with i = r;) than in the FF (i.e. at large r, wherejEij! 0;r! +1, with i = r;). This is in agreement with the consequence that NF detectionallows to probe higher spatial periodicities of the sample prole, i.e. provides higher resolution.Furthermore, the FF (i.e. both eld components for r ! +1)vanishes at  =0;more rapidlythan at  = =2, while the NF (i.e. both eld components for r ! 0)diverges at  =0;morerapidly than at  = =2; that is, the FF is zero at the poles and maximum on the equatorial plane,corresponding to the classic polar diagram of radiation intensity I / cos2 , while the oppositeholds for the NF (see Chapter 5). It is the same as if close to the dipole the two mismatchedcharges originating it can be distinguished, at whose positions the eld actually diverges. Thisexample shows how far can be the NF and the FF behavior of the same sample.The question still left open here is where a boundary surface r = r0 between the twoNFandFFdomains can be traced, stating that for r >r0 the FF character prevails more and more, whilefor r<r0 the same holds for the NF character. This can be for example determined by lookingagain at Eqs. 1.14-1.16 describing the electric eld of the oscillating dipole. For each of the twonon-zero components it is possible to set a borderline between FF and NF. After the dependenceon the dierentpowers of r, one can dene the contributions 1 and kr inside the parentheses asthe NF [28] and the FF term, respectively1. Remember that for the second term the factor ijust means that a phase shift of =2 occurs (as i = ei=2). The transversal component E of theelectric eld also contains a term -k2r2, that is obviously even more dominating in the FF (i.e. forlarge r). Nevertheless, let us neglect this term for the moment. The boundary at which the NFand the FF contributions are equal can thus be dened as 1=kr, i.e.:rr0 = 1k = 2 ' 0:159 (1.17)For r>rr0 the dominating character of the eld will be FF (e.g. 80% eld intensity provided bythe FF for r =2rr0), while for r  rr0 the eld will be mainly NF (e.g. 80% eld intensity providedby the NF for r =0:5 rr0).
1In fact, this is just one of the possible choices. In another common view it is only the kr2 term, called radiationterm, whichisidentied with the FF, while the term 1 describing the NF is also called Förster eld [29, 30], andthe intermediate kr term is not considered at all in the two limits.
6 1 NEARFIELD OPTICSThe extra term k2r2 in E tells us that the boundary limits rr0 and r0 in the two directions !rand ! are not the same, i.e. the boundary surface is not a sphere. For an exact analysis of thelimit in the transversal direction ! , the modulus of the NF term 1 and of the FF term ikr k2r2have to be compared. When this is done, the condition1=ikr0  (kr0)2 (1.18)brings to the solution: r0 ' 0:125 (1.19)Therefore, for the NF along the transversal direction to be dominating on the FF a closer positionhas to be tested, in agreement with the shape of the NFFF boundary domain as sketched inRef. [31]. Hence,tostay on the safe side one can take this limit as the distance belowwhich theNF is dominating for all the directions of the eld. When the range 300-800 nm is consideredfor the visible spectrum, it is possible to see that such limit r0 varies in the range '40-100 nm,respectively.1.4 Neareld optical microscopy, principles and congura-tionsThe general idea behind NF optical microscopyisthat an illuminating, propagating light waveis partly scattered byasubwavelength size sample feature (or tip) into an evanescent light eldconned close to its source, containing high spatial frequency information concerning the sample.This is then partly scattered again into a propagating wave directed toward a detector bymeansof another subwavelength size object, that is the tip (or a sample feature). These twoschemescorrespond to illumination and collection mode, respectively [32]. Moreover, both schemes canwork either in transmission or reection, depending of the path covered by light coming to thesource to the sample and nally to the detector, i.e. if source and detector lie on the same sideof the sample, or on the opposite side. Just from these basic considerations it is clear that avariety of NF optical microscope setups are possible, as depicted in Fig. 1.1. The reason for thisis that, thanks to the relatively high penetration of photons inside matter as compared to thatof electrons, the three parts of the microscope, namely source of radiation, detector, and sample,can be easily split and re-combined in a number of allowed geometries. Furthermore, the use ofNF enables one more level of dierentiation, between allowed and forbidden (i.e. evanescent) lightmodes. On the other hand, despite the fact that in all the worldwide spread laboratories no twoidentical home-built NF optical microscopes exist, the basic principles are the same (as suggestedat the beginning of the paragraph) and many setups are somehow equivalent.From the point of view of the involved physics, more than the geometry of the microscope compo-nents, the NF optical microscopes can be better distinguished into two families. Most instrumentsmakeuseof a metal coated optical ber tip, with a hole in the coating at the tip apex, of sev-eral tens of nanometers in diameter. This hole forms the aperture required for superresolution,following the original idea of Synge [18] and according to the rst historical NF optical micro-scope realization by Pohl [19] and Lewis [20]. Such microscopes belong to the so-called familyof aperture NF optical microscopy (or aSNOM [33]). Corresponding to the illumination andcollection schemes mentioned above, in these instruments the metal coated tip can be used bothas a nanoscale light source, thus providing local illumination of just a small area of the sample, oras a nanodetector, collecting in turn the light irradiated on the sample by an ordinary FF lightsource. The electromagnetic eld is squeezed in the ber tip taper, whose borders act as a metallicwaveguide smaller than the diraction limit. A trade-o applies to these tips: for resolution to be
1.4 NEARFIELD OPTICAL MICROSCOPY, PRINCIPLES AND CONFIGURATIONS 7
TIR
collection
r
e
f
l
e
c
t
i
o
n
transmission
s
c
a
t
t
e
r
i
n
g
allowed
f
o
r
b
i
d
d
e
n
t
r
a
n
s
m
i
s
s
i
o
n
,
transmission,
illum. coll.
r
e
f
l
e
c
t
i
o
n
(c)
(a) (b)Figure 1.1: (a) aperture SNOM; (b) apertureless SNOM; (c) PSTM, as a special case of (b).maximized one needs a small aperture hole. On the other hand, this limits the throughput of theber, such that a small transmission is obtained for small holes.The approach of the Photon Scanning Tunneling Microscope (PSTM) [34,35] is quite dierent.This setup belongs to the family of so-called apertureless SNOM. In this case the tip works just asa scattering center placed at its pointed extremity, and the evanescent eld is therefore scatteredinto propagating waves, which are collected in several dierentways. Most commonly collectionoccurs along the optical ber itself, towards the opposite end of the tip, but it can also be operatedwith a FF detector placed nearby the scattering region. In this case the tip can be made not ofglass, at the free end of a tapered optical ber, but out of any sharpened body, either a metal likefor a STM tip or an insulator (ceramic) like for an AFM cantilever tip, for example.The instrument used during the presentwork is a PSTM. As the name itself suggests, the basicprinciple of operation of PSTM is quite similar to that of STM [16]. Namely, a tunneling occursthrough a gap in which the involved particles are photons instead of electrons. As a consequence,some properties of the instrument can be derived in a straightforward manner from the STM. Oneimportantpoint in common is that the signal across the gap decays exponentially with the gapwidth. However, the decaylengthismuch larger in this case: if for STM it is of the order of 1 nmor smaller, for the PSTM is at least 100 times longer, of the order of several hundreds nm (seeEq. 1.27). For the same reason, if one neglects the theoretical resolution limit due to diraction,the practical lateral resolution of the PSTM is also more limited than that of the STM. Indeed,due to the weaker dependence of the signal on the tipsample distance, a portion of the tip that islarger in all dimensions (and thus also parallel to the sample substrate) interacts with the samplein the PSTM, determining the signal intensity whichispicked up. Other points of contact betweenNF optics by PSTM and electronic states of matter show up when the surface plasmon polaritonmodes of metallic samples are investigated, see Chapter 2. Concepts of solid state physics havealso been used for photons, like Blochwaves, Brillouin zones and band gaps, in the study ofphotonic crystals (see e.g. Ref. [2,36]). Even if of no direct interest for the presentwork, theseanalogies will come back in the PSTM investigations of samples consisting of regular arrays ofmetal structures (see Chapters 6, 7). Nevertheless, a fundamental dierence with the STM is thatan electron in a potential can be described as a scalar, while a photon whichbelongstoawaveisavectorial entity, and the direction of propagation of the incident exciting wave has to be takeninto account. In fact, for a better understanding of the PSTM it is more useful to develop furtherthe point of view of the electric eld of a lightwave, as presented in the theoretical discussion ofthe next Section.
8 1 NEARFIELD OPTICS1.4.1 Total internal reection: evanescent eld intensities and decayAs suggested in Fig. 1.1(c), in the PSTM the illumination of the sample is provided by an extendedincident beam under conditions of total internal reection (TIR). The reason for TIR is thatthe light background above the sample substrate is thus greatly suppressed, allowing an easierdiscrimination of the signal of interest in the lightintensity collected by the optical ber tip. Thesituation is discussed in detail in Fig. 1.2, where an incident beam consisting of plane, harmonicwaves is impinging on a glassair interface.
n
2
E
TE
0
x
z
-y
n
1
k
in
E
0
TM
α
Figure 1.2: General situation of a wave incident on the glassair interface inside a prism in TIRcondition. The electric eld can always be split into a TE-like (ETE0 ) and a TM-like (ETM0 )component, perpendicular and parallel to the plane of incidence, respectively.The incidenteldisdescribedby the following expression:!Ei(!r;t)=!E 0 ei(!k i
!r !t) (1.20)where t is the time and !r apoint(x; y; z) in space, and the constantvector !E 0 contains bothamplitude and polarization informations. !k i is the wavevector and ! the angular frequency of theincidentwave. In general, it will have a TM and a TE component, that will be indicated as !ETM0and !ETE0 , respectively. The Fresnel equations tells us about the solution of the electromagneticproblem at the glassair interface, concerning both cases of TM and TE polarization, and thereforeany cased of mixed polarization. As we will exploit only the evanescent eld on the air sideunder TIR conditions, we focus our attention on the transmitted eld equations, and on angles ofincidence larger than the critical angle for TIR, c. The equations describing the components ofthe transmitted eld !E t are [25]:Etx = ETE0 2n1 cospn21  n22 ez=ÆA ; (1.21)Ety = ETM0 2cosqsin2  n221qn421 cos2 +sin2  n221 ez=ÆA ; (1.22)Etz = ETM0 2 cos sinqn421 cos2 + sin2  n221 ez=ÆA (1.23)
1.4 NEARFIELD OPTICAL MICROSCOPY, PRINCIPLES AND CONFIGURATIONS 9where n1 and n2 are the refractive indexes of medium 1 (glass) and 2 (air), respectively, whilen21 = n2=n1 is the relative refractive index of 2 with respect to 1. The eld components are alsoshifted in phase with respect to the incident eld, as described by the following expressions:x = arctan qn21 sin2  n22n1 cos ; (1.24)y =z  2 ; (1.25)z = n1qn21 sin2  n22n22 cos (1.26)
(b)
(a)Figure 1.3: Transmitted electric eld amplitudes (a) and intensities (b) for TIR, as a function ofthe angle of incidence . Both quantities are normalized to either the total electric eld amplitude(or intensity), or to the eld amplitude (or intensity) concerning the considered component only.Refractive indexes: n1=1.51, n2=1.
In Fig. 1.3 the behavior of the TIR transmitted eld components and the respectiveintensitiesrightbeyond the interface (z=0 i.e. ez=ÆA=1) are shown, as a function of the angle of incidence .Both cases of TM and TE polarization are considered. The plotted quantities are normalized toeither the total electric eld amplitude (or intensity), or to the eld amplitude (or intensity)concerning the considered componentonly. For TE polarization this makes no dierence, as onlyone eld component is present, but this is not the case for TM polarization. In particular, inthe latter case the value of the NF enhancement G is inuenced. This quantity is dened inPSTM images as the ratio of the maximum detected NF intensityabove the sample of structuresover the intensityofthelight eld incident on the structures (see Chapter 5). As in the PSTMthe experimentally observed enhancementisalways based of the total measured intensity,inTMpolarization this measured G should be multiplied by a correction factor GTMcorr as plotted in (b),in order to obtain a measure of the selectiveintensity enhancement for the considered component.
10 1 NEARFIELD OPTICSThe evanescent eld decay length ÆA, corresponding to an attenuation of a factor 1/e with pene-tration in the second medium along z; is:ÆA = 2qn21 sin2  n22 (1.27)where the index A recalls that this is the decay length for the eld amplitude, while for the eldintensity I jEj2 the corresponding 1/e decay occurs at ÆI = ÆA=2.Fig. 1.4 reports the evanescent eld intensity decay length as a function of the angle of incidenceand the wavelength in the visible, ÆI(; 0), for a TIR glassair interface.
( )
λ
δ
i
α
(nm)
(nm)Figure 1.4: Evanescent eld intensity decay length ÆI as a function of the angle of incidence and the lightwavelength , for a TIR glassair interface as shown in Fig. 1.2. Refractive indexes:n1=1.51, n2=1, (values see Appendix A).
In TIR the transmitted wave, which is evanescent along the z direction perpendicular to thesubstrate, is propagating in medium 2 along the plane of the 12 interface in a direction given bythe projection of the incidentwavevector on this plane, !k ik. Due to the condition of equal phasefor the waves at interface 12 at the pointofcontact with it, one has:
(!k 
 !r  !t)=0 (1.28)with t time, ! angular frequency, !k wavevector and !r position on the plane of the interface.In particular, at !r =0 in TIR it turns out that for incident and transmitted wave !it = !tt,such that it is possible to conclude !i = !t = !. Due to this and to the fact that !r belongs tothe plane of the 12 interface, the above condition Eq. 1.28 is simplied to 
!k k =0. As all thewavevectors lye in the same plane (plane of incidence) and in TIR no reected beam exists, thisreduces to: ktk = kik = ki sin (1.29)for the condition connecting incident and transmitted wavevector.
1.4 NEARFIELD OPTICAL MICROSCOPY, PRINCIPLES AND CONFIGURATIONS 11From the expression of the wavevector as a function of the wavelength, one obtains:2eff = 21 sin (1.30)where wehave called eff the eectivewavelength of the incidentwave running above the substrate(medium 2), and 1 the wavelength below it (medium 1). As ! constant means that the ratioci=i (speed of lightover wavelength) is the same in the two media i=1,2, and the speed of lightin matter is ci = c0=ni with c0 the speed in vacuum and ni refractive index of the consideredmedium, wehave: eff = 0ng sin (1.31)(where it has been further assumed that medium 1isglassand2isair, at it is indeed in mostcases of practical interest, so that n1 = ng and n2 =1). In practice eff and 0 are very close(exactly at the critical angle of incidence c for onset of TIR it is ng sinc=1), but in generalit is eff .0 (as  & c). In Appendix B a list of values of eff as a function of the angleof incidence  are reported. As a parameter, the incident vacuum wavelength 0 is consideredto vary among several dierent values of interest in our PSTM experiments, corresponding toavailable laser sources. In fact, for a matter of practical convenience not eff but eff/2 is listedin the table, as this quantity is in rst approximation the expected periodicity observed in form ofinterference fringes in our PSTM images, due to the inplane backscattering of the incidentwaveby the sample structures (see Chapter 5).
12 1 NEARFIELD OPTICS
1.5 ConclusionIn this chapter a brief introduction to the optical neareld (NF) has been given, starting fromthe possibilitytoovercome the resolution limit (Abbe barrier) aecting conventional fareld (FF)optical microscopy. As an example making clear some striking dierences between the NF andthe FF behavior, the eld prole of an oscillating dipole has been shortly discussed rst. Whileone could think this to be quite a special case of little interest, in fact our plasmon resonant goldnanoparticles, which constitute the sample of interest for the PSTM investigations, are indeedworking as oscillating dipoles, to a rst approximation (see Chapter 2). After a rough denitionof the NF region and a short description of the possible NF optical microscopy congurations, wefocused on the TIR conditions applied in our instrument, i.e. the PSTM. The formulas for eldamplitudes and evanescent eld exponential decay length were provided. This is the foundationrequired to understand the practical implementation of our PSTM setup, as presented in Chapter 3.