A transfer function is an important notion in anyimaging technique, which allows one to relate animage and an object presuming, of course, that theprocess of image formation is linear. Most often, theintensity distribution of light in an image is relatedto the intensity distribution at the correspondingobject plane. It determines the spatial resolution ofan imaging instrument and also allows for consid-eration of the inverse problem. In conventional (far-field) microscopy, knowledge of the optical transferfunction makes it possible to reconstruct the objectstructure from a recorded image.1 In scanning near-field optical microscopy (SNOM), multiple scatter-ing in the probe–sample system usually takes place.Therefore one should consider the self-consistentfield, which greatly complicates the situation. It has
been shown,2 however, that the probe–sample in-teraction is negligible in photon tunneling (or, ingeneral, collection) SNOM (PT-SNOM) with un-coated fiber probes. For this reason the transferfunction is mainly discussed in relation to the col-lection SNOM.
The notion of a transfer function in near-fieldmicroscopy is not a new one. It had been implicitlyassumed already in 1993 by Tsai et al.3 and byMeixner et al.4 in their works with experimentaldetermination of spatial resolution of the PT-SNOM. Theoretical investigations, showing thatthe signal detected in the PT-SNOM is proportionalto the square modulus of the electric near field5 andsuggesting accounting for a finite size of the probeby means of a transfer function,6 resulted in wideusage of the intensity transfer function (ITF) whiletreating the experimental results.7–11 This ITF re-lates the Fourier transform (FT) of a near-field op-tical image to the FT of a corresponding intensitydistribution. Nevertheless, the very existence ofsuch a transfer function has been questioned judgedon the basis of numerical results,12 experimentallyin the context of probe characterization13 and the-oretically14 using treatment of image formation de-veloped previously by Greffet and Carminati.15
Moreover, it has been shown that the notion of ITFcan be used in only some particular cases, i.e., using
I. P. Radko ([email protected]) and S. I. Bozhevolnyi are withthe Department of Physics and Nanotechnology, Aalborg Univer-sity, Skjernvej 4C, DK-9220 Aalborg ost, Denmark. N. Gregersenis with the Department of Communications, Optics, and Materials,NanoDTU, Technical University of Denmark, Building 345V, DK-2800 Kongens Lyngby, Denmark.
Received 3 August 2005; revised 29 December 2005; accepted 9January 2006; posted 13 January 2006 (Doc. ID 63884).
several approximations.14 Instead, an amplitude-coupling function (ACF) has been suggested for thecharacterization of the collection SNOM. On theother hand, it is rather difficult to measure both themagnitude and the phase of the ACF for a givenSNOM arrangement, since it requires the use ofphase detection techniques. From the point of viewof SNOM resolution, the most important character-istic is the ACF magnitude or, more precisely, itsdecay for high spatial frequencies corresponding toevanescent field components. It is this decay that(along with the noise level) determines the spatialresolution attainable with the SNOM. For this rea-son, we are concerned in this paper with the mea-surements and simulations of the ACF magnitude,especially in the domain of evanescent field compo-nents.
The paper is organized as follows. In Section 2 wereview the main formulas, explaining the image for-mation process, and introduce a simple approxima-tion for the ACF based on the idea of point-dipoledetection by an extremity of a SNOM fiber tip.5 Then,in Section 3, we present our measurements of theACF magnitude in different configurations, detectingfar- and near-field wave components. Section 4 pre-sents the results of numerical simulations of the cou-pling efficiency in two-dimensional (2D) geometry. Ithas been argued elsewhere16,17 that the use of 2Dsimulations simplifies considerably the computa-tional effort without loss of the essential features ofthe SNOM imaging process. In Section 5 we discussthe results obtained utilizing the suggested approxi-mation for the probe ACF. In Section 6, we offer ourconclusions.
2. Background
Let us consider image formation in the collectionSNOM, in which a fiber tip scanned near an illumi-nated sample surface is used to probe an optical fieldformed at the surface (Fig. 1). This field is scatteredby the tip, also inside the fiber tip itself, couplingthereby to fiber modes that are formed far from theprobe tip and propagating toward the other end of thefiber to be detected.14,15 To relate the fiber mode am-plitudes to the plane-wave components of the probedfield, it is convenient to make a plane-wave Fourierdecomposition of the incident electric field E�r�, z� inthe plane z � 0 parallel to the surface and passingthrough the probe tip end:
E�r�, 0� �1
4�2 ��F�k�, 0�exp�ik� · r��dk�, (1)
where r� � �x, y� and z � 0 are coordinates of the tipend, k� � �kx, ky� is the in-plane projection of the wavevector, and F�k�, z� is the vector amplitude of thecorresponding plane-wave component of the incidentfield. Since, as already mentioned, the multiple probe–sample scattering can be disregarded in the collectionSNOM, the field described by Eq. (1) is considered to be
the only one that makes a contribution to the detectedsignal. To relate the plane-wave components to thefiber mode amplitudes excited in the probe fiber, wefurther assume that the fiber used is single mode andweakly guiding, oriented perpendicular to the surface,and terminated with a probe tip possessing axial sym-metry. Note that, for weakly guiding fibers (having avery small index difference between the fiber core andits cladding), the guided modes represent appro-ximately transverse-electromagnetic (TEM) waves.Thus, in the considered case, any propagating fielddistribution can be decomposed into two orthogonallypolarized fiber modes, whose amplitudes we shall de-note as Ax and Ay. It should be mentioned in passingthat the origin of the probed field [see Eq. (1)] isirrelevant in this consideration and that the samplesurface serves merely as a reference plane surfacesustaining the field with high spatial frequencies(i.e., evanescent wave components).
We can further reason that, in the case ofs-polarized illumination, one should expect to find, forsymmetry reasons, only a y-polarized component ofthe field in the above plane-wave decomposition anda similarly polarized fiber mode (Fig. 1). In the case ofp polarization, one should, however, expect to find x-and z-polarized field components in the above decom-position. Again using symmetry considerations, ittranspires that both components can contribute onlyto the x component of the fiber mode, since thez-polarized field component having the symmetricorientation with respect to the probe axis does prop-agate along the x axis. Taking these arguments into
Fig. 1. (Color online) Coordinate system and schematics of near-field detection by a fiber tip. Point S situated at distance z0 from thetip extremity represents an effective detection point at which cou-pling of the incident field to the fiber mode is considered to takeplace.
account we can express the mode amplitudes of a(single-mode) fiber as follows:
Ay�r�� �1
4�2 ��Hyy�k��Fy�k�, 0�exp�ik� · r��dk�,
Ax�r�� �1
4�2 �� �Hxx�k��Fx�k�, 0�
� Hxz�k��Fz�k�, 0��exp�ik� · r��dk�, (2)
where Hij�k�� are the coupling coefficients that ac-count for the contribution of the jth plane-wave com-ponent to the ith component of the fiber mode field.The matrix H composed of (amplitude-coupling) co-efficients Hij plays a role in the ACF, which repre-sents the coupling efficiency for various spatialfrequencies and, for example, determines the maxi-mum attainable spatial resolution when imagingwith the SNOM.
The exact form of an ACF depends, in general, onthe probe characteristics (shape, refractive indexcomposition, etc.) and can be rather cumbersome todetermine.15 It is clear, however, that the ACF mag-
nitude should decrease for high spatial frequencies.This is probably the most important feature of theACF that should be present in any ACF approxima-tion. We suggest a description of the ACF using thefollowing model. Let us view the detection process asradiation scattering by a pointlike dipole situatedinside the tip (point S in Fig. 1) at the distance z0 fromthe tip extremity. Its scattering efficiency (into agiven fiber mode) depends only on the correspondingcomponent of the (incident) field at the site of thedipole:
Ay�r�� � c�Ey�r�, z0�,
Ax�r�� � c�Ex�r�, z0� � c�Ez�r�, z0�. (3)
Making decompositions in the field expressions abovethat are similar to that in Eq. (1) but at the planez � z0 containing the detection point S, one obtains
Fig. 2. (Color online) (a) Fiber tip used in both far- and near-field measurements. (b) Experimental setup for the far-field measurementsin collection mode along with (c) the results obtained for s (solid curve) and p polarization (dashed curve).
Finally, taking into account that Hxx � Hyy and con-sidering only evanescent field components, the fol-lowing relations for the ACF components can bewritten:
Hxx�k� � k0� � Hyy�k� � k0� � c� exp��z0�k�2 � k0
2�,
Hxz�k� � k0� � c� exp��z0�k�2 � k0
2�, (5)
where k0 is the wavenumber in air, i.e., k0 � 2��� and� is the light wavelength in air. The obtained (ap-proximate) ACF expressions [Eqs. (2) and (5)] re-present the main result of our consideration. Thisapproximation being simple and rather transparentalso preserves the vectorial character of the ACF14,15
and contains the detection roll-off for high spatialfrequencies. We demonstrate below that it can also befitted reasonably well to approximate both experi-mental and simulation data.
3. Experimental Results
A. Far-Field Measurements
In this part of the experiment, first, a laser(� � 633 nm, P 1.5 mW) beam having s or p polar-ization (the electric field is parallel or perpendicularto the plane of incidence) illuminated the tip of a fiberprobe under investigation [Fig. 2(a)] under differentangles of incidence [Fig. 2(b)]. The power of coupledradiation was registered at the output end of the fiberby a photodetector. The probe was produced from asingle-mode optical fiber (the cutoff wavelength was780 nm). On the scale of the probe tip apex, wherethe coupling actually takes place, the (Gaussian) la-ser beam can be considered as a plane wave, so thedetected signal should be proportional to the squaredmagnitude of the corresponding ACF component, atleast in the case of s polarization. Measurements forboth polarizations showed that the detected signaloscillates strongly with the spatial frequency for rel-atively small angles of incidence ��40°� and rapidly
Fig. 3. (Color online) (a) Schematics of a fiber tip used in illumination mode and the light distribution in front of the fiber tip producedby a CCD camera. (b) Far-field signal dependencies measured in illumination mode for s (solid curve) and p polarization (dashed curve).
Fig. 4. (Color online) (a) Schematics of near-field measurements in collection mode: P, polarizer; F, focusing lens; M, rotational mirror tochange angle � and thereby angle �, which was kept larger than the critical angle of total internal reflection. (b) Signal dependenciesobtained for s (solid curve) and p polarization (dashed curve). Fitting shown by thin solid lines is made for each curve as explained in Section5.
decreases for large angles [Fig. 2(c)]. We considerstrong oscillations as the manifestation of Mie(shape) resonances that can be excited in an uncoatedfiber tip by propagating waves. Perhaps a better un-derstanding of such behavior of the registered signalcan be perceived considering the image of light inten-sity distribution obtained with a CCD camera placedin front of the probe, with the laser radiation beingcoupled in the fiber from another end [Fig 3(a)]. Here,one should expect to find the reciprocity between theillumination and collection SNOM modes.15 The ex-perimental dependence of the detected throughputillumination on the angle of detection is shown in Fig.3(b) and is produced in the setup, similar to thatshown in Fig. 2(b), with the laser and the photode-tector being exchanged. A similar dependence can beobtained by making a (properly chosen) cross sectionof the image from Fig. 3(a). The fact that this crosssection should be displaced away from the fiber axiscan be explained by a slight displacement of the ro-tational plane of the photodetector (or the laser) withrespect to the fiber probe axis.
B. Near-Field Measurements
The near-field measurements provided more accu-rate and straightforward data, allowing us to char-acterize the probe detection efficiency in the domainof evanescent field components. Our experimentalsetup is schematically shown in Fig. 4(a). A SNOMapparatus was used to scan a small area of theprism surface illuminated from the side of the prismwith a slightly focused laser beam (� � 543.5 nm,P 1.5 mW) being totally internally reflected. Bychanging the incident angle � and using a high-refractive-index prism �n � 1.73�, it was possible torealize the illumination with spatial frequencies
varying from 1.0 up to 1.67, when being expressed ina normalized form, i.e., k��k0 � n sin �. The resultsobtained for s and p polarizations are shown in Fig.4(b) by bold solid and dashed curves, respectively. Toseparate the graphs, the data for s polarization havebeen moved up by a factor of 3. Also in this case, wecan reason that the measured signal dependenciesare directly related to the squared ACF magnitudes.
4. Numerical Simulations
We also calculated the electromagnetic field distribu-tion around the fiber tip subject to the illuminationfrom below. For an incoming plane wave with a spe-cific value of k� � |k�|, we determined the amountof power coupled to the fundamental fiber mode.Varying the value of k�, we obtained the (spatial-)frequency-dependent transmission, which is againrelated to the squared magnitudes of the ACF com-ponents.
Because full vectorial calculations of light scatter-ing on three-dimensional (3D) nanoscale structuresare computationally demanding, we have considereda 2D fiber with uniformity along one lateral axis. Thegeometry of the refractive index profile is depicted inFig. 5. The core and cladding indices of the fiber are1.459 and 1.457, respectively; the core diameter is4 m, and, to increase computation speed, a reducedcladding diameter of 20 m was chosen. The openingangles were varied from 10° to 70°.
The simulation method used to calculate the elec-tromagnetic field was the eigenmode expansiontechnique. In this procedure, the refractive indexgeometry is divided into layers of a uniform refrac-tive index profile along a propagation axis, herechosen as the Z axis. Eigenmode and propagationconstants are calculated in each layer, and thefields at each side of a layer interface are connectedusing the transfer-matrix formalism. The generalmethod is described in Ref. 18. For the eigenmodecomputation, a plane-wave basis was chosen with adiscrete number of plane waves; further details ofthe eigenmode determination are given in Ref. 19.
The graded part of the fiber tip was modeled usinga discrete step-index profile, and the entire geometrywas enclosed in a box with periodic boundary condi-tions. The number of steps in the step-index profileand plane waves in the discrete basis as well as thebox size were increased until the transfer functionconverged. The resulting transmissions for two polar-izations of the incident field and for various openingangles are depicted in Fig. 6 as a function of normal-ized spatial frequency. It is seen that, for propagatingfield components, the transmission exhibits strongoscillations, whereas it rapidly and monotonously de-creases for high spatial frequencies corresponding toevanescent field components. This behavior is quali-tatively similar to that observed in the experimentreported above.
5. Discussion
First, we should notice a resemblance in the angu-lar dependencies of the signal detected by the
Fig. 5. (Color online) Two-dimensional geometry used in numer-ical simulations of the coupling process. The following parametershave been used in the simulations: � � 633 nm, core diameter of 4�m, core index of 1.459, cladding index of 1.457.
SNOM probe [Fig. 2(c)] and simulated (for the prop-agating field components) transmission (Fig. 6):Both are of oscillatory behavior featuring maximaand minima. In the domain of the evanescent fieldcomponents �k��k0 1�, it transpires from both ex-perimental and theoretical results that the ACF roll-off is well pronounced, implying the possibility ofusing our approximation [Eqs. (5)]. In the case of spolarization (containing evanescent wave compo-nents), the first formula in Eqs. (5) can be applieddirectly. For p polarization, however, one has to con-sider two components, parallel and perpendicular tothe surface plane �Ep � E� � E��, whose detectionefficiencies related to the coefficients Hxx and Hxz aredifferent. When deriving an expression for the ACF inthis case, one should take into account the ratioEz�Ex � ik���k�
2 � k02�1�2, which can be obtained easily
from the Gauss law: div E � 0. Furthermore,one should expect that the coupling coefficients�Hxx and Hxz� are different not only in magnitude butalso in phase, arriving at the following expressions:
Hs�k� � k0� � A exp��z0�k�2 � k0
2�,
Hp�k� � k0� � B�1 � �k0�k��2 exp��z0�k�2 � k0
2�� C exp�i��exp��z0�k�
2 � k02�. (6)
We fitted the experimental signal dependencies mea-sured for the illumination with evanescent field com-ponents with the squared functions from Eqs. (6),since the experimental results represent power mea-surements. Considering s and p polarizations, wehave obtained the following fitting parameters:z0 140 nm, B�C 1.2, and � �. It is seen that thecorrespondence between experimental and approxi-mated dependencies is reasonably good [Fig. 4(b)]and that the fitting parameters are consistent withthe observations reported previously. Thus a good
correspondence between calculated field intensitydistributions at a 100 nm distance over a rough nano-structured surface and near-field intensity distribu-tions measured with the tip–surface distance of5 nm was found,20 suggesting that the effective de-tection point was located inside a fiber tip at a dis-tance of 100 nm from its extremity. Measurementsand simulations of field phase and amplitude distri-butions over diffraction gratings also supported theconcept of effective detection occurring at some dis-tance from the tip extremity.21 In addition, it hasbeen experimentally demonstrated that the SNOMdetection introduces polarization filtering, so that thefield component perpendicular to the surface is de-tected (with an uncoated fiber tip) less efficientlythan the parallel component.22 In our case, the polar-
Fig. 6. (Color online) Numerical results for the coupled (transmitted through the fiber) electric field amplitude obtained for (a) s and (b)p polarization and for different apex angles � (Fig. 5): 10° (dashed curve), 30° (dotted curve), 50° (dashed–dotted curve), and 70°(dashed–dotted–dotted curve). Note that in the evanescent field domain (k��k0 � 1) the data are multiplied by a factor of 500 for spolarization (all angles) and p polarization (for 10° and 30°) and by a factor of 2000 for p polarization and for 50° and 70°.
Table 1. Fitting Parameters for Different Tip Apex Anglesa
aParameters found from fitting the data obtained in the numer-ical simulations of detection efficiency for different spatial frequen-cies: distance z0, position of the effective detection point of anear-field probe; �, phase difference between two field componentsin the p-polarized incident light; A, contribution coefficient fromthe incident field of the s-polarized light to the coupled field; B,contribution coefficient from the in-plane incident-field componentof the p-polarized light to the coupled field; B�C, ratio of the con-tributions from the in-plane and perpendicular-to-the-planeincident-field components of the p-polarized light to the coupledfield. The values are presented for different tip apex angles of thenear-field probe.
ization filtering is seen in the fact that the ratio B�Cwas found to be larger than 1.
The same fitting procedure has been carried outusing the simulated data corresponding to the eva-nescent field components for all considered tip apexangles (Table 1). Here the agreement between exact(calculated) and approximated [Eqs. (6)] dependen-cies is seen to be rather good (Fig. 7). Use of the fittingparameters allows one to interpret the results of nu-merical simulations revealing their connections tothe features observed experimentally. Thus the in-crease in the distance z0 with the apex angle impliesthat sharper tips are better suited for SNOM imagingwith high spatial resolution, which has been one ofthe first features associated with the PT-SNOM.3–5
On the other hand, the increase in coefficients A andB with the apex angle means that the detected signalis expected to be larger for blunter tips, a feature thathas also been frequently noted (e.g., Refs. 5 and 13).Note that the ratio B�C that is very large for sharptips decreases rapidly with the increase in the apexangle, a tendency that agrees well with the experi-mental results on polarization filtering22 and on de-tection of surface plasmon polaritons, whose electricfield is predominantly polarized perpendicular to thesurface.23 Moreover, the fitting parameters obtainedfor the simulated (in 2D geometry) data (Table 1) arealso consistent with those found when fitting the ex-perimental signal dependencies measured with thefiber tip with an apex angle of 22° [Fig. 2(a)]. Fi-nally, the phase difference �� �� between two con-tributions to the signal for p polarization can beaccounted for partly with the phase difference be-tween the corresponding field components, i.e., Ez
and Ex, as was seen above, and partly with the phasedifference in the dipole scattered radiation along andperpendicular to the incident field polarization. How-ever, this explanation should be elaborated (whichrequires further investigations) to obtain a better un-derstanding of the SNOM detection and its polariza-tion properties.
Concluding this section, we comment on the non-
monotonic behavior of simulated signal dependenciesfor p polarization and the apex angles of 30° and 50°[Fig. 7(b)]. It can be qualitatively understood in thefollowing way. Coupling of the perpendicular (to thesurface) field component �Ez� is less efficient than thatof the parallel one �Ex�, whereas, for low spatial fre-quencies k��k0 ��1�, the former is much stronger thanthe latter for p-polarized radiation as was seen above.When increasing the spatial frequency k��k0, the par-allel component increases rapidly and, being detectedmore efficiently, comes into force. The transition fromthe signal that originated mainly from the perpen-dicular field component �Ez� to that related mainly tothe parallel one �Ex� results in local minima in thetransmission curves shown in Fig. 7(b) for the tipapex angles of 30° and 50°. Note that it remains to beseen to what extent this effect found here with 2Dsimulations would manifest itself in the experimentsand accurate (3D) modeling.
6. Conclusion
Summarizing, we have considered the characteriza-tion of a near-field optical probe in terms of detectionefficiency of different spatial frequencies paying spe-cial attention to the detection of evanescent field com-ponents. Experimental results as well as numericalsimulations carried out in 2D geometry have beenpresented for the detection of both s and p polariza-tion and found consistent. The detection roll-off forhigh spatial frequencies corresponding to the evanes-cent field components was both observed in the ex-periment and obtained during the simulations of theincident field. It has been further fitted using a sim-ple expression for the transfer function, which wasderived introducing an effective point of (dipolelike)detection inside the probe tip. We have found it pos-sible to fit reasonably well both the experimental andthe simulation data for evanescent field components,implying that the developed approximation of thenear-field transfer function can serve as a simple,rational, and sufficiently reliable means of fiber probecharacterization.
Fig. 7. (Color online) Evanescent field domain of numerical results shown in Fig. 6 along with the fitted (as explained in Section 5) curvesshown with solid lines. Marking of the curves simulated for different apex angles is as in Fig. 6.
We believe that the results obtained serve as abetter understanding of the SNOM imaging and canbe used for treating experimentally obtained images,especially those containing significantly differentspatial frequencies (e.g., corresponding to differentscaling objects), by taking explicitly into account thedependence of detection efficiency on the spatial fre-quency [see Eqs. (5) and (6)]. In principle, by measur-ing the detected signal at one spatial frequency(corresponding to the evanescent illumination), oneshould be able to predict fairly accurately the level ofthe signal at other (higher) spatial frequencies and,e.g., determine the attainable spatial resolution for agiven noise level. The relatively simple expressionsthat we derived can also be further elaborated, serv-ing as the first approximation so that the very shapeof the fiber tip apex (e.g., the tip radius) would betaken into account.
This research has been carried out within the frame-work of the Center for Micro-Optical Structures sup-ported by the Danish Ministry for Science, Technologyand Innovation, contract 2202-603�40001-97.
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