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Pseudo modulation transfer function in reflection scanning near-field optical microscopy

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Page 1: Pseudo modulation transfer function in reflection scanning near-field optical microscopy

1 September 1998

Ž .Optics Communications 154 1998 167–172

Pseudo modulation transfer functionin reflection scanning near-field optical microscopy

Dominique Barchiesi 1

Laboratoire d’Optique P.M. Duffieux, Unite Mixte de Recherche au Centre National de la Recherche Scientifique 6603,´Institut des Microtechniques de Franche-Comte, UniÕersite de Franche-Comte, F-25030 Besancon Cedex, France´ ´ ´

Received 16 February 1998; revised 6 May 1998; accepted 2 June 1998

Abstract

In near-field optical microscopy, the resolution is strongly related to the experimental conditions of illumination and tothe distance between the tip and the sample. Therefore, an intrinsic modulation transfer function is generally meaningless innear-field microscopes. In this paper, we show that experimental microscopes can be characterized by a linear transfer,according to a few experimental conditions, we consequently establish the existence of a pseudo modulation transfer

Ž . Ž .function PMTF . We calculate it, in the R-SNOM reflection scanning near-field optical microscope case. The PMTF is anefficient tool for determining the SNOM capabilities, if the tip to sample distance is large enough, without resorting towell-known test samples. This characterization could provide information on the resolving power of the microscope,working on a wide scanning zone, to choose an interesting sample area. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 07.79.Fc; 42.30.Lr; 42.30.Va; 42.30.KqKeywords: Near-field; Image processing; Resolution; Microscopy; Scanning near-field optical microscopy; Modulation transfer function

1. Introduction

Nowadays, super-resolution is currently reached by var-Ž . w xious near-field optical microscopes SNOM 1 . The reso-

lution is currently determined by visual inspection of thew xexperimental images 2–4 . Only a few papers have been

concerned with the methods for calculating the resolutionin experimental near-field images. Some are related to

w x w xstatistical considerations 5 , transfer function of the tip 4 ,w x w xlocal resolution 6 , properties of near-field 7 or signal to

w xnoise ratio in hybrid microscopes 8 . However, as far aswe know, no study has been devoted to the characteriza-tion of the SNOM, when the tip is scanned in CHMŽ .constant height mode , more than 100 nm from the sur-face. This imaging mode is sometimes used to identify a

1 E-mail: [email protected]

worth studying zone of the sample, before imaging it withhigh resolution power, with smallest tip to sample dis-tance. This imaging mode avoids tip crashing on thesample and seems to be a better method to obtain a pure

w xoptical signal 9 . The problem is to determine the ultimatestructure that can be imaged in this mode. Thus thecharacterization of the set-up, working in CHM mode, isproposed, using the concept of pseudo modulation transfer

Ž .function PMTF . The experimental needs are indicated fordefining an ultimate resolution of near-field microscopes,working in CHM, in a preliminary imaging process. Weverify that theses assumptions can be made in particularexperimental images. We thus characterize the experimen-tal setup, by using these images. In the next sections, for

Žclarication we will use the ad hoc notations QNF quasi-. Ž .near-field and QFF quasi-far-field to discern the two

images that are recorded respectively at dslr4"lr30and ds4lr5"lr30.

0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0030-4018 98 00316-2

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( )D. BarchiesirOptics Communications 154 1998 167–172168

2. Concept of pseudo modulation transfer function( )PMTF in scanning near-field optical microscopy( )SNOM

The purpose of the PMTF is the experimental charac-terization of the R-SNOM, working in constant distancemode, in the quasi-near-field or in the quasi-far-field zoneŽ .Fig. 1 . The R-SNOM apparatus uses the same bareoptical fiber tip to illuminate the sample and to detect the

Ž .optical signal see Fig. 1 . Actually, the CHM imagingmode enables the scanning of an unknown sample, withoutany risk of tip crash. This is a first common workingmode, to detect an interesting zone of the sample thatcould be imaged later, with a shorter tip to sample separa-tion. The problem is to know, in this first approach, whatis the smallest detail, that can be imaged, otherwise, someinteresting objects could be obliterated in further scanning.

The lateral resolution depends strongly on the setup butit has been shown that the resolution in the near-fieldimages depends mainly on the polarization and on the

w xdistance between the tip and the sample 10 . Actually byconsidering the diffraction of the incoming light by asample, it is well known that two types of waves arediffracted: propagating waves and evanescent ones. Thedetection of evanescent waves enables the increase of thelateral resolution in the images, but the amplitude of theevanescent waves decreases rapidly versus the distancefrom the sample. Another behavior could a priori compli-cate the concept of transfer function in near-field optics.When the tip scans the sample in constant height modeŽ . w xCHM 9,7 , the distance between the tip and the sample

Fig. 1. The R-SNOM apparatus, working in constant height modeŽ .CHM . The distances between the tip and the sample for thequasi-far-field zone and the quasi-near-field zone are specified.

varies. Consequently, the capability of obtaining high reso-lution could vary as a function of the lateral position of the

w xtip. Nevertheless, Hecht et al. 9 showed that the constantheight mode was the best way to obtain pure opticalinformation, without any artifact related to z-motion of the

Ž .probe Fig. 1 .If the distance between the tip and the sample is a few

nanometers, a transfer function cannot be defined a prioriwithout strong approximations, except in the case of very

w xlow sample reliefs 11 . On the contrary, if the distancebetween the tip and the sample is greater than lr4, we

Ž .show that a pseudo-modulation transfer function PMTFcan be defined in R-SNOM, working in the CHM mode, ifsome experimental conditions are checked. Like in classi-cal microscopy, the PMTF will provide the ultimate resolu-

w xtion that can be achieved. Kann et al. 12 demonstratedtheoretically that the influence of the tip on the dielectricsample images can be neglected if the distance betweenthe tip and the sample is greater than lr10.

To our knowledge it is the first time that the experi-mental PMTF was evaluated in the R-SNOM experimentalcase, working in constant height mode, with a non-coatedtip. The concept of transfer function was theoretically

w xdiscussed in near-field, by Courjon 13 , Carminati andw x w xGreffet 11 and Kann et al. 12 . The main problems are

underlined in these papers.– The experimental transfer function should be defined

by a ratio between two experimental data, that is why it isŽ .called pseudo modulation transfer function PMTF . The

SNOM images are effectively dependent on the topogra-Ž .phy or on the optical index inhomogeneities and on

Ž .conditions of illumination polarization, etc. . Actually, weuse the near-field data as reference, and we calculate thePMTF by using the quasi near-field data of the samesample. We use the invariance of the PMTF by translationto test its existence. The sample is made of dielectric,cylindrical SiO dots on a flat Si substrate. The lateral size2

of the dots is 1 mm and the distance between the dots is 3mm.

w x– The near-field image formation is complex 14 andinvolves a linear transfer only if the roughness of thesample is much smaller than the wavelength and the

w xinverse of detected spatial frequency 15,16 . In the experi-mental images that we processed, the ratio of the height of

Ž . Žthe dots 150 nm to the wavelength is lower than 0.2 the.wavelength is 789 nm . We use the fast Fourier transform

Ž .FFT of the samples of the scanning lines. It is assumedthat the ratio of the height to the inverse of the Fourier

Žspatial frequency is lower than 1 the scanning step is 55.nm .– In the same way, the tip has to be considered as a

passive probe. This assumption is acceptable since the tipw xand the sample are dielectric materials 17–19 . Moreover,

the interaction between the tip and the sample rapidlydecreases with their distance d, being negligible for d)

w xlr10 typically, even if the tip is metal coated 12 . In the

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( )D. BarchiesirOptics Communications 154 1998 167–172 169

processed images, d is respectively equal to lr4"lr30and 4lr5"lr30. The near-field data are recorded in thequasi near-field zone with the dielectric tip.

– Another experimental parameter enables the exis-tence of the PMTF: the lateral size of the tip is close to the

Ž .scanning increment f50 nm . Size or shape effects of thetip on the images can then be neglected in the imaging

w xprocess 20 . We will verify this fact with the standarddeviation of the resolution in the last section.

3. Calculation of the PMTF

The PMTF is defined as the ratio of the Fourier spec-trum of a segment of a scanning line to the associatedFourier spectrum of the corresponding scanned line whenthe tip is closer. The purpose of this calculation is toinvestigate a cut-off frequency, like in classical mi-croscopy except that the image formation is so complexthat we cannot consider the sample geometrical profile as a

reference. The use of scanning line segments allows thedetermination of the local resolution and the method ischecked by using three images recorded at different scan-ning distances.

Let us detail the method we developed to calculate thePMTF. The PMTF is the ratio of the Fourier spectrum ofthe considered data to the associated Fourier spectrum of

Ž .the near-field data Fig. 2 . This point shuns the choice ofa typical Dirac object that could help to define the transferfunction. On the other hand, no discussion of the imageformation related to all experimental parameters is neces-sary. To remove artifacts due to the possible piezo decaybetween scanning lines, we extract part of the scanninglines, leading to a ‘‘local PMTF’’, in the very scanning

Ž .line. Any scanning line is 128 pixels long 7 mm and weextract successively some samples that are 30, 40, 60 and80 pixels long and we calculate the associated PMTF. Toavoid the aliasing effect, each sample is joined to its ownflipped. The junction is smoothed to avoid any artifactual

Ž .high resolution sps0.01, 5 pixels . The next step con-sists in calculating the PMTF and in smoothing the result

Fig. 2. Three scanned lines of the same sample extracted from the raw maps on the left, that are recorded in the ‘‘quasi-far-field’’ zoneŽ . Ž .QFF: ds4lr5"lr30 , in the ‘‘quasi-near-field’’ zone QNF: dslr4"lr30 and in the near-field zone df20 nm. The pixelspacing is 55 nm. The resolution is f150 nm in the NF data, f200 nm in the QNF data and f250 nm in the QFF data.

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( )D. BarchiesirOptics Communications 154 1998 167–172170

Ž .by a cubic spline fitting with smoothing factor sps0.001 .For sps0, it is the least-squares straight line and forsps1, it is the cubic spline interpolation. A value of spbetween 0.0005 and 0.01 works as well. These limitsweakly depend on any sample particularities. The smooth-ing facilitates the determination of the location of the firstminimum of the PMTF that corresponds to the limit ofresolution. At the same location, the phase of the PMTFexhibits the first local extremum. Before this extremum,the variations of the phase is monotonic, whereas the phasevaries rapidly after this first extremum. Therefore, wecalculate the limit of resolution, corresponding to this

Žcutoff frequency the first minimum of the PMTF is.chosen as a pessimistic criterion .

The property of invariance by lateral translation of thePMTF is verified, if the result is the same for various linesof the same image, for various subareas of the same lineand for various images recorded with the same tip tosample separation. On the other hand, the cut-off fre-quency is strongly dependent on the longitudinal transla-

Ž .tion d . In our case, since the sample is not homogeneous,we use the results obtained on various lines and we provethat the cutoff frequency of the PMTF is around the samelocation. Actually, we show that the cutoff frequency isweakly dependent on the sample profile, therefore, thesetup can be characterized.

Fig. 2 shows three scanned lines of the same sampleŽ .extracted from the gray level maps on the left . The maps

Žare recorded in the ‘‘quasi-far-field’’ zone QFF: ds. Ž4lr5"lr30 , in the ‘‘quasi-near-field’’ zone QNF: ds

. Žlr4"lr30 and in the near-field zone somewhere the.tip rubs . The three scanned lines correspond to the same

Žzone of the sample. By visual inspection scanning edgesw x.method 12 , one can note that the resolution can be

related to the slope of the edges of the images of the dots:f150 nm in NF data, 200 nm in QNF data and 250 nm inQFF data. The near-field data contain high spatial frequen-cies and thus they can be used as reference for the PMTFcalculations. A histogram of the Fourier spectrum could

w xshow this. A previous study 8 of the same setup workingwith shear-force feedback gave a resolution better than 180nm in near-field data. One can note that the near-fieldimage exhibits only the edges of the dot instead of theprofile of the dot itself. These well-known phenomena

Ž w xhave been theoretically studied high pass filtering 10 orw x.field enhancement 7,14 . Therefore, the dot is more

precisely located in the NF data. Its lateral size is actuallyŽabout 1 mm discussions on the whole images can bew xfound in Ref. 7 , the comparison with theory shows that

.the probe can be considered as passive .Fig. 3 shows the PMTFs calculated for three equally

distributed samples of 40 pixels as a function of thegeneralized Fourier frequency. The samples are extractedfrom the lines in Fig. 2. The generalized Fourier frequencyp is independent of the number of pixels or of the realc

size of the sample. The Shannon frequency corresponds to

Fig. 3. The PMTF is calculated with three equally distributed 40pixel samples, as function of the generalized Fourier frequencyp . They are extracted from the scanned lines of Fig. 2. The firstc

minimum of the PMTF indicates the cutoff frequency. The phasevariations of the PMTF show a monotonic variation up to thecutoff frequency. After the cutoff, the phase varies rapidly.

p s0.5 corresponding to two pixels. The spline fitting ofc

the PMTF diminishes the possible perturbation due to thenoise level in the raw data. The fitting and the noise in thenear-field image implies the second maximum of thePMTF, located at p f0.4. The noise implies strong varia-c

tions in the PMTF after the cutoff, for high spatial frequen-cies. The spline fitting in this zone, with a small splineparameter involves this non-significative behavior, relatedto the shape of the fitted PMTF before the cutoff. The

Ž .strong variations of the phase fitted by the spline showthe same behavior. Therefore the resolution in nanometers

Žcan be obtained simply by dividing the scanning step 7.mmr128 by p . In QFF, the cutoff frequency is close toc

0.21, leading to a resolution rf7000r128r0.21f260nm. In QNF, the cutoff frequency is close to 0.27, leadingto resolution r f 7000r128r0.27 f 200 nm. Conse-quently, the PMTF can be evaluated in the QFF and in theQNF zone. The whole image can be processed. The ‘‘lo-

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( )D. BarchiesirOptics Communications 154 1998 167–172 171

Fig. 4. The histograms of the cutoff generalized Fourier frequency p are calculated with 5 equally distributed samples, 30 pixels long, incŽ .each line of the whole image 128=128, 7 mm . The spline factor is 0.001.

cal’’ resolution seems to be more scattered in the dottedcurves with the same value of spline parameter. We use a

Ž .statistical study on various scanning lines Fig. 4 to verifythe property of invariance by translation of the PMTF. Fig.4 shows the histograms of the positions of the cutoff

Ž .frequencies p in the PMTF. The resolution r nm can bec

calculated by using the scanning step ss7000r128 nm,Ž .independently of the extracted sample: r nm ssrp .c

Table 1 shows the mean resolution and the standarddeviation obtained in the overall image, by extractingsamples that are 80, 60, 40 and 30 long. This study provesthat PMTF can be defined in QFF and in QFF data,leading to a characterization of the setup. The dispersion ofthe results is more important in the QNF image than in theQFF one. The results are more sensitive to the spline factorthat could be smoothly diminished to get more aggregated

Table 1The resolution r is expressed in nm and the standard deviation s

is indicated. The spline factor is equal to 0.001

Ž . Ž .Samples r nm QNF s r nm QFF s

30 pix. 215 55 230 6040 pix. 195 45 230 7060 pix. 180 40 210 6080 pix. 182 37 222 65

results. It has to be noticed that the standard deviation isaround the lateral size of the tip end and the scanning step,therefore the dispersion could not be significantly reduced.

4. Conclusion

The R-SNOM setup can reach a resolution of 160 nmw x Ž .8 , by near-field scanning NF data . This value wasobtained with the same apparatus, working in constantdistance mode with shear-force feedback and imaginganother test sample. In this paper, we characterize thesame R-SNOM set-up, working in constant height mode,with a greater separation between the tip and the sample.For this purpose, we showed that the pseudo modulationtransfer function can be used and that it is an efficient toolin the quasi-near-field zone of scanning. The resolutionlimit is obtained by calculating the cutoff frequency. Anapplication of the method to images recorded by anothernear-field microscopes, like STOM data, is in progress.The experimental calculation of the pseudo modulationtransfer function will enable the improvement of the exper-imental setup, and will enable to define a priori the tip tosample separation to reach a given resolution. We demon-strated, with a pessimistic criterion, that the R-SNOMsetup, using a bare tip, can reach a resolution better thanlr3, by working in constant height mode with a tip to

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( )D. BarchiesirOptics Communications 154 1998 167–172172

sample separation equal to lr4. On the contrary, asŽexpected, in the quasi-far-field zone the tip is about 4lr5

.away from the sample , the method gives a result that isclose to the Rayleigh limit.

Acknowledgements

I am grateful to Michel Spajer and to Olivier Bergossifor having developed the experimental setup and for pro-viding experimental images. I thank Daniel Courjon forfruitful discussions on the problem of modulation transferfunction in near-field microscopies and for the improve-ment of this paper.

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