---
title: "Investigations of the interference of surface plasmons on rough silver surface by scanning plasmon near-field microscope"
authors: ["V.N. Konopsky", "K.E. Kouyanov", "N.N. Novikova"]
affiliation: "Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow region, 142190, Russia"
journal: "Ultramicroscopy"
year: 2001
volume: "88"
pages: "127-138"
doi: "10.1016/S0304-3991(01)00045-6"
type: journal-article
site_group: "Scanning plasmon near-field microscopy"
url_abstract: ""
url_pdf: "https://valery.konopsky.com/additional_pdf/Investigations of the interference of surface plasmons on rough silver surface by scanning plasmon near-field microscope.pdf"
language: en
source_tex: ""
source_pdf: ""
---
## Abstract

A scanning plasmon near-field microscope with gold and silver tips,
operating in tapping mode of atomic force microscope (AFM) is used to
measure the distribution of the near-field intensity of surface plasmons
on rough silver surfaces. Using the fast Fourier transformation (FFT) of
near-field images it is shown that the distribution of the near-field
intensity on the surface is the result of the interference between
scattering plasmons and the initial plasmon beam. Multiple scattering
effects such as backscattering enhancement of surface plasmons are also
observed. It is shown that a nonuniformity in the registration of the
scattered light leads to some artifacts in near-field images. Several
registration modes of the light signal are considered and it is shown
that recording the light signal at the second harmonic of the tapping
frequency one can pick out the signal associated with an electromagnetic
(em) resonance in a tip-surface (sphere-plane) structure. Possible
implementations of this em resonance for studies of local permittivities
and local nonlinear susceptibilities of intermediate media between the
tip and surface with a subtip resolution are discussed.

*Keywords: Surface plasmons, Apertureless scanning near-field.*

> **Abstract.** A scanning plasmon near-field microscope with gold and
> silver tips, operating in tapping mode of atomic force microscope
> (AFM) is used to measure the distribution of the near-field intensity
> of surface plasmons on rough silver surfaces. Using the fast Fourier
> transformation (FFT) of near-field images it is shown that the
> distribution of the near-field intensity on the surface is the result
> of the interference between scattering plasmons and the initial
> plasmon beam. Multiple scattering effects such as backscattering
> enhancement of surface plasmons are also observed. It is shown that a
> nonuniformity in the registration of the scattered light leads to some
> artifacts in near-field images. Several registration modes of the
> light signal are considered and it is shown that recording the light
> signal at the second harmonic of the tapping frequency one can pick
> out the signal associated with an electromagnetic (em) resonance in a
> tip-surface (sphere-plane) structure. Possible implementations of this
> em resonance for studies of local permittivities and local nonlinear
> susceptibilities of intermediate media between the tip and surface
> with a subtip resolution are discussed.

PACS: 07.79.Fc; 61.16.Ch; 73.20.Mf\
</div>

# Introduction

In a scanning near-field optical microscope (SNOM) a high resolution
optical image of a sample is obtained by scanning a subwavelength light
source or detector close to the sample surface . Aperture-type probes,
e.g., aluminum-coated optical fibers tapered at their end have an
aperture radius of typically 30–50 nm. In an apertureless scanning
near-field microscope (aSNOM) a sharp tip of a scanning tunneling
microscope (STM) or AFM is illuminated from the outside by an external
light source and scatters the evanescent field which is then far-field
detected . The lateral resolution of aSNOM is determined by the tip size
and reaches the value of 17–20 nm .

A scanning plasmon near-field microscope (SPNM)  may be considered as an
aSNOM in which a traveling surface plasmon field on the silver surface
is used as an external excitation source of aSNOM tip. Using gold and
silver coated cantilever tips it is possible to excite an em resonance
in the tip-surface (sphere-plane) structure and reach a resolution less
than a tip radius . In this paper we use this approach to study the
behavior of surface plasmons on rough silver surfaces.

The plan of this paper is as follows: in section 2 we describe our
experimental setup and different registration modes of the light signal.
In section 3 we present and discuss our experimental results: a
distance-dependent variation of the light signal, the distribution of
the near-field intensity of the surface plasmons on rough silver surface
and FFT of this intensity, consider artifacts in near-field images due
to a nonuniformity in the registration of the scattered light and lastly
describe multiple scattering effects. In section 4 we give the
conclusions.

# Experimental setup

Our experimental setup is schematically shown on Figure 1. A beam of a
cw He-Ne laser ($\lambda=632.8$ nm, $I\simeq 1$ mW) incident on a
silver film at a defined angle $\theta_0$ of total internal reflection
excites the surface plasmons at the silver-air interface (it is usual
Kretschmann configuration). Silver films of thickness 500$\AA$ were
prepared by thermal evaporation of Ag on the base of the quartz prism at
the pressure $5\cdot 10^{-6}$ Torr at a room temperature. The
deposition rate of silver was about 5 Å/sec. The thickness of silver
films was measured using a quartz crystal monitor disposed near the base
of the quartz prism. For studies of multiple scattering effects we have
need for silver films with large pointlike surface irregularities, and
even with several closely spaced irregularities (to investigate a
backscattering enhancement effect). For this purpose we proceed as
follows: after completion of a work with a silver film and removing one
from the prism base by nitric acid we evaporate the next film on the
prism base without preliminary polishing and/or fire polishing of the
prism base (as it done if smooth film is desirable). After two – four
such turns we obtain the films which may be used in studies of multiple
scattering effects.

The excitation of the SPs is recognized as a minimum in the reflected
laser intensity and can be understood as destructive interference
between the light reflected from the silver-air and the silver-quartz
interfaces . A commercial scanning probe microscope "Solver P-47" of
"NT-MDT" firm  with gold and silver coated silicon cantilevers has been
used in common AFM mode (a bend of the cantilever is held constant by
AFM feedback during scan) and in tapping mode of AFM (an amplitude of
the cantilever vibration is held constant by AFM feedback during scan).

Several registration modes of the light signal are used in our setup.\
1) *"Internal reflection registration mode"* — it is the registration of
the intensity variation of the reflected light beam. (scattered light
rays (S1–S2) are removed by the set of diaphragms – SoD). This
registration mode is the one that was used in the works .\
2) *"Internal scattering registration mode"* — it is the registration of
the intensity of the conical light radiation arisen from an elastic
scattering of the surface plasmons in the angle $\Delta\varphi$
between rays S1 and S2 (reflected light beam is removed by the set of
diaphragms – SoD). S1 and S2 are the extreme left and extreme right rays
of the conical scattered light radiation that is collected by the lens
on the photodiode. This registration mode is similar to the one that was
used in the works , but in these works the overall conical scattered
light radiation was collected by a cylindrical mirror (i.e.
$\Delta\varphi =2\pi$ in these works; in our case
$\Delta\varphi\simeq 23^0$).\
3) *"External scattering registration mode"* — it is the registration of
the plasmon-photon scattering using the multimode fiber with diameter
300 $\mu$m and numerical aperture $\approx 0.4$ placed in the
immediate vicinity (0.5 mm) of the cantilever tip. An undesirable
scattered light from an AFM laser diode ($\lambda\simeq
670$ nm) has been removed by a double monochromator. The light signal
has been detected by a photo-multiplier tube (PMT) placed on the exit of
the double monochromator.

In each mentioned above registration modes the light signal can be
recorded:\
a) *without modulation*: that is AFM operates in the common mode and the
light signal detected at zero frequency;\
b) *at the first harmonic of the cantilever vibration*: that is AFM
operates in the tapping mode and the light signal detected at a
frequency $\Omega$ of a cantilever vibration;\
c) *at the second harmonic of the cantilever vibration*: that is AFM
operates in the tapping mode and the light signal detected at a
frequency $2\Omega$.

An typical value of the resonant frequency of gold- and silver-coated
cantilevers in our setup was $\Omega\approx
40$ kHz. An amplitude of the cantilever vibration was about 150 nm, and
it was kept constant by AFM feedback during the scan.

# Experimental results and discussion

## Distance-dependent variation of the light signal and the em resonance in a tip-surface structure

To understand the advantages of the light registration at the second
harmonic of the tip-surface distance modulation one must consider the
variation of the light signal while the tip-sample distance is varied.
The distance-dependent intensity of the light signal in the "external
scattering registration mode" is presented on the Figure 2. A
gold-coated silicon tip with $R\simeq 190$ nm is used to obtained this
dependence. One can see that light intensity decreases during the tip
withdrawal from the surface, but some maximum ("bump") takes place at
the curve at $x=35$ nm.

Similar "bumps" on distance-dependent intensity curves were also
detected by other authors in different registration modes of SPNM. See,
for example,  and  ( "internal reflection registration mode") and also 
( "internal scattering registration mode"), but we have found that in
the "external scattering registration mode" this maximum is more
pronounced. An detailed discussion about the origin of this maximum has
been done elsewhere . Here we present only our main conclusions. We
believe that this maximum takes place due to the em resonance in the
tip-surface (sphere-plane) structure. The em resonance in the
sphere-plane structure was thoroughly studied in the early 80s in the
context of a light emission from small particle tunnel junctions , and
in the context of surface-enhanced Raman scattering . In visible region
this em resonance is of greater intensity when the tip and the surface
are noble metals. The resonances in the sphere-plane structure at
$d<<R$ are often called "gap modes". Eigen frequencies of these modes
may be determined from the next approximated analytical expression :
``` math
\begin{equation}
{\epsilon_0\over\epsilon_1'(\omega)}+
{\epsilon_0\over\epsilon_2'(\omega)}=
-\left(n+{1\over 2}\right)\sqrt{2d\over R}\; , \quad
n=0,1,2,\dots\; ,

\end{equation}
```
where $\epsilon_2'(\omega)$ and $\epsilon_1'(\omega)$ are the real
parts of the permittivities of the sphere and the plane. In our
configuration (see Figure 3) $\epsilon_2'(\omega)$ and
$\epsilon_1'(\omega)$ are the real parts of the permittivities of a
tip and a surface, $R$ is a radius of curvature of the tip and $d$
is a distance between the tip and the surface. Therefore if we will
excite the gap between the needle and the surface by an em radiation
with a fixed frequency (e.g. by He-Ne laser) the em resonance in this
sphere-plane structure occurs at the next "resonance" distance between
the tip and the surface:
``` math
\begin{equation}
d_{\mathrm res}=2R\left[{\epsilon_0\over\epsilon_1'
(\omega_{_{\mathrm HeNe}})}+ {\epsilon_0\over\epsilon_2'
(\omega_{_{\mathrm HeNe}})}\right]^2

\end{equation}
```
(hereinafter we shall consider "gap mode" with $n=0$). In other words
an "effective" dipole moment of the sphere-plane structure has a maximum
amplitude at $d=
d_{\mathrm res}$, and it decreases when the tip approaches to the
surface at the distance $d< d_{\mathrm res}$ and moves away from the
surface at the distance $d> d_{\mathrm res}$. For good agreement with
observed value of $d_{\mathrm res}\simeq 35$ nm it is necessary that
"effective" permittivity of intermediate layers (chiefly silver sulfide
tarnish layer : $\varepsilon'_{\mathrm Ag_2S}=8.7$ and adsorbed water:
$\varepsilon'_{\mathrm H_20}=1.8$) between the tip and the silver
surface was about $\epsilon_0\approx 2$.

The important characteristic feature of the sphere-plane em resonance is
the lateral dimension $L$ of the em field localization between the
sphere and the plane. It is approximately equal :
``` math
\begin{equation}
L\approx\sqrt{2dR}\; .

\end{equation}
```
Therefore at $d<<R$ this dimension is less than the tip radius
($L<R$). As a result the huge increase of the light intensity under
sphere takes place .

If we will modulate the tip-surface distance at a frequency $\Omega$,
then we can pick out the signal associated with the sphere-plane
resonance by the detecting a light signal at the frequency $2\Omega$.
Returning to Figure 2 one can see on insets in dashed-line frames the
elucidation of the mechanism of the $2\Omega$ frequency generation at
the tip-surface distance modulation by $\Omega$ frequency: one may
consider how the light intensity changes while the tip to make one
complete cycle of vibration. Points 1–5 on the intensity variation curve
correspond to the points 1–5 on the tip position curve. It is seen from
these curves that the light intensity turns twice while the tip makes
one turn.

The registration of the light signal at the second harmonic of the
distance modulation (that is, in fact, the registration of the em
resonance signal) opens up several new possibilities for studies of
surface with nanometer resolution. The most straightforward of them is
the use of huge field enhancement of the em field under tip for
registration of the nonlinear effects such as Raman scattering or second
harmonic generation with a resolution about $L$. For example, the
registration of the second harmonic will provide the information about
nonlinear susceptibilities of the $$tip-intermediate layer-surface$$
structure with a resolution even better than $L$. Another possible
implementation of the tip-surface em resonance opens up if one will
detect (in each point of the surface) the *spacing* between the tip and
the surface at which the em resonance occurs. So this distance strongly
depends on the intermediate media permittivity ($d_{\mathrm res}\sim
\epsilon_0^2$) such a registration will provide the information about
$\epsilon_0$ with a subtip resolution. For example, molecules with a
resonance line at the laser frequency deposited on the surface may be
visualized in such a manner.

## Distribution of the near-field intensity of the surface plasmons on rough silver surface: interference between scattering plasmons and the initial plasmon beam

Figure 4b illustrates the distribution of the near-field intensity of
the surface plasmon field on a silver surface (direction of the initial
surface plasmon beam designated by the arrows). Scan size is
15800 nm$\times$<!-- -->16800 nm. The optical signal was recorded at
$2\Omega$ frequency in the "internal reflection registration mode"
(i.e. $1c$ mode in our notation, see
section (<a href="#setup" data-reference-type="ref" data-reference="setup">2</a>)).
A gold-coated silicon tip with $R\simeq 190$ nm is used to obtained
this image. From mathematical data processing of the topography of this
part of the surface (Figure 4a) we obtain the root-mean-square height of
the surface roughness $\delta=<(\Delta z)^2>^{1/2}\simeq 3.8$ nm.

From comparison of the surface topography (Figure 4a) and the near-field
image (Figure 4b) one can make the next conclusions:\
a) prominent surface hillocks, such as the hillocks 1–5, appear in the
near-field image as the white spots 1–5 (reasons why hillocks have
negative contrast have been done elsewhere )\
b) other prominent surface features, such as the ditch 6 in the up-right
corner of the Figure 4a, also lead to a expected interference pattern in
the vicinity of these features,\
but\
c) apart from that, some additional ripples and fringes, which cannot be
associated with any particular surface features, are appearing on the
near-field images.

The similar near-field pictures have been observed by other authors
using SPNM (see, for instance ) and SNOM . But to our knowledge, the
reasons of appearance of such "cluster structure" of the em field which
not associated with any particular features in the surface topography
have not been clearly illuminated. To attain such an understanding one
must perform the Fourier transformation (FFT) of the near-field image.
On the Figure 4d one can see a distinct circle in the FFT of the
near-field image (the second circle arises because of inherent feature
of Fourier transformations: a FFT image is symmetric respective to the
origin of the coordinates). The explanation of the appearance of such a
circle in the FFT image is given on the Figure 4c. The initial surface
plasmon beam is scattered by surface irregularities and interference
between the initial surface plasmon beam and scattering plasmons takes
place. At elastic scattering the wavevector of the initial plasmon
${\mathbf k}_{\mathrm pl}$ changes its direction, but its magnitude
remains the same ($|{\mathbf k}_{\mathrm
pl}|=|{\mathbf k}_{\mathrm sc}|$). The distribution of the SP
near-field intensity on the surface is $I_{\mathrm pl}+
I_{\mathrm sc}+
2\sqrt{I_{\mathrm pl}I_{\mathrm sc}}
\cos\left(({\mathbf k}_{\mathrm pl}-
{\mathbf k}_{\mathrm sc})
{\mathbf r}\right)$. So the "vectors of the interference gratings"
${\mathbf K}_{\mathrm int}={\mathbf
k}_{\mathrm pl}-{\mathbf k}_{\mathrm sc}$ are distributed so that their
ends lie on the circle with radius equals
$|{\mathbf k}_{\mathrm pl}|$. One can see from the FFT image that
forward scattering (small ${\mathbf K}_{\mathrm int}$) is of the
greatest intensity. In the case when effects of the multiple scattering
(see <a href="#multi" data-reference-type="ref"
data-reference="multi">3.4</a>) are negligible the intensity along this
circle must be proportional to the roughness function
$\delta h({\mathbf g}_r)$ (i.e. FFT of the surface topography). The
angular dependence of surface plasmon scattering by surface
irregularities is of great concern, for example, in studies of a laser
damage of metal mirrors. The surface plasmons play a crucial role in
such a damage  and elimination of the surface plasmons caused by the
surface roughness is of considerable importance for increasing the laser
damage threshold of metal mirrors .

##  Artifacts in near-field images due to nonuniformity at the registration of the scattered light

In some works with SPNM, operating in the "internal scattering
registration mode", the large scale interference pattern is silhouetted
against of the "cluster structure" background (see, e.g., ). We believe
that this effect occurs due to a nonuniformity in the registration of
the scattered light in this mode. To prove this assumption we shut off
not only the reflected beam, but also the scattered light rays between
the reflection beam and S2 beam in our setup (see Figure 1). That is
only scattered light rays between the reflected beam and S1 beam have
been used in the "internal scattering registration mode". The results
are presented in Figure 5 (scan size is
7200 nm$\times$<!-- -->7200 nm). A gold-coated silicon tip with
$R\simeq 190$ nm is used to obtained this image. One can see that a
large scale interference pattern is appeared in near-field images
recorded by different methods: Figure 5c — *without modulation*,
Figure 5d — *at the first harmonic of the cantilever vibration*,
Figure 5e — *at the second harmonic of the cantilever vibration*. In the
FFT image (5f) of Figure 5e a pronounced point, which is corresponded to
this large-scale interference pattern, is appeared (the second point is
symmetric to the first one respective to the origin of the coordinates).
The reasons of this are as follows: inasmuch as the forward scattering
is of the greatest intensity, an appearance of a particular "preferable"
direction in the registration system (i.e. nonuniformity) leads to the
enhancement of the contribution from plasmons that scattered in this
particular direction. The period of this interference pattern is
``` math
\begin{equation}
\Lambda={\lambda\over 2\sin({\Delta\varphi\over 2})}\; ,

\end{equation}
```
where $\Delta\varphi$ is the mean angle between the direction of the
scattered light rays and the direction of the initial plasmon beam.

From Figure 5 one can also see that near-field images recorded at the
second harmonic of the cantilever vibration has the best resolution.

##  Multiple scattering effects

### SP scattering between the tip and surface irregularities

In the works done with tapered fibers in near-field microscopy (see  and
bibliography mentioned therein) the use of a dielectric tip assures that
the interaction between the tip and SPs is negligible and presence of
the dielectric tip does not influent on the near-field images. In our
experimental arrangement we, in fact, use a large dipole moment of the
tip-surface structure as a local probe of the SP field distribution. On
the flat surface, where the dipole $p$ of the surface-tip structure is
the only prominent scattered center, intensity of SP scattering in the
"internal scattering mode" and a value of the SP extinction (scattering
plus absorbtion) induced by this probe dipole in the "internal
reflection mode" are proportional of SP intensity in the absence of the
tip ($p=\alpha E_{\mathrm sp}$, $I_{\mathrm
sc}\sim \ddot p^2\sim |E_{\mathrm sp}|^2$, where $\alpha$ – the
polarizability of the tip-surface structure).

The situation changes when other effective scattering centers of SPs are
present on the surface. In this case the SP scattered by the tip may be
rescattered by such a scattering center and returns to the tip. That is
the SP field at the tip position becomes not the same as in the absence
of the tip. This type of effects of multiple scattering becomes
important for very large surface hillocks — when a cross-section of
surface plasmon scattering of a large hillock and the one of the tip
become comparable in magnitude. In this case typical V-shaped figures
near large point-like hillocks occur in the SPNM image. The detailed
description of such effects can be found in . We have found that in our
setup these effects became well-defined at the surface hillock height
$h>100$ nm. But even in this case the circle in the FFT of the
near-field images takes place, but the intensity distribution along the
circle changes. Figure 6 is the illustration of this effect. Scan size
is 25000 nm$\times$<!-- -->20000 nm, the root-mean-square height of
the surface roughness $\delta=<(\Delta z)^2>^{1/2}\simeq 13.2$ nm. The
near-field image is recorded in "internal reflection registration mode"
at the second harmonic of the cantilever vibration. A silver-coated
silicon tip with $R\simeq 150$ nm is used to obtained this image.

### Backscattering enhancement effect

Another effect of multiple scattering is a backscattering enhancement of
surface plasmons on a random surface. For a detailed consideration of
this subject one can see the theoretical works , and experimental work 
in this area of exploration (although in mentioned above works
significantly different experimental procedure was considered). We shall
confine our account to a purely qualitative discussion of the problem.
The initial plasmon wave scatters from a point on the rough surface and
reach another point on the surface where it is scattered once again. The
time-reversed version of this process also will occur, in which the
initial plasmon wave interacts with the second point and then couples to
an outgoing wave at the first point. These two sequences produce diffuse
scattering contributions that interfere constructively in the opposite
direction to the initial plasmon wave direction. Further, this argument
holds for all pairs of points on rough surface and, upon averaging, a
narrow backscattering peak is produced. Therefore
${\mathbf k}_{\mathrm 2sc}=-{\mathbf k}_{\mathrm pl}$ and in FFT image
at ${\mathbf K}_{\mathrm int}\equiv 2{\mathbf
k}_{\mathrm pl}$ a local maximum associated with the backscattering
enhancement will occur.

The effect of the backscattering enhancement is easy to record on such a
part of a surface where several closely spaced prominent (not
necessarily very large) hillocks are present. The example is done on
Figure 7. Scan size is 18000 nm$\times$<!-- -->18000 nm, the
root-mean-square height of the surface roughness $\delta=<(\Delta
z)^2>^{1/2}\simeq 7.2$ nm. The optical signal is recorded in "internal
reflection registration mode" at the first harmonic of the cantilever
vibration. A silver-coated silicon tip with $R\simeq 150$ nm is used
to obtained this image. The maximum arises from the backscattering
enhancement denotes by a dotted arrow on the FFT image (Figure 7c).

It may be noted that authors of the works  tried to observe the effect
of the backscattering enhancement by looking for the interference
fringes with $\lambda_{\mathrm sp}/2$ period in the direction of the
initial SP beam. It has been pointed out  that desirable interference
pattern is difficult to observe due to the presence of other
interference patterns related to different scattered SPs.

We hope that the present work demonstrates that the Fourier
transformation of the near-field images is a simple and powerful tool
for studies of such effects.

#  Conclusions

In this paper we have used a scanning plasmon near-field microscope,
operating in tapping mode of AFM, to measure the distribution of the
near-field intensity of surface plasmons on rough silver surfaces. Using
the fast Fourier transformation (FFT) of near-field images it has been
shown that distribution of the near-field intensity on the surface is
the result of the interference between scattering plasmons and the
initial plasmon beam. Multiple scattering effects such as backscattering
enhancement of the surface plasmons have been also observed using FFT of
near-field intensity. We have shown that a nonuniformity in the
registration of the scattered light leads to certain artifacts in
near-field images.

We have used gold and silver-coated cantilevers to enhance an
electromagnetic resonance in a tip-surface (sphere-plane) structure. We
have recorded the optical signal at the second harmonic of tapping
frequency to pick out the signal associated with the sphere-plane em
resonance. The spatial resolution of the light signal at the tip-surface
em resonance is determined by the dimension of the light field
localization beneath the tip and it may be less than the tip radius.
Possible implementations of this em resonance for studies of local
permittivities and local nonlinear susceptibilities of intermediate
media between the tip and surface with a subtip resolution have been
discussed.

In closing it may be said that proposed method of the investigation of
the surface plasmon distribution on the rough surface may be improved in
several aspects. One of them is next: to improve the resolution of the
FFT images one must perform the large-size scans
(100 $\mu$m$\times$<!-- -->100 $\mu$m or more) of the surface. In
this case one can reach the ultimate FFT resolution
$\delta k/k\sim\lambda/l_{\mathrm sp}$, where $l_{\mathrm
sp}$ – mean free path of surface plasmons. Several interesting fine
effects may be observed in this case, for example, the energy band gap
of the surface plasmon curve due to interaction of the initial plasmon
beam (${\mathbf k}_{\mathrm pl}$) with spatial frequency component of
the surface roughness
${\mathbf g}_{\mathrm r}\simeq 2{\mathbf k}_{\mathrm pl}$, when an
amplitude of this component is not very small (for more details see  and
references therein). On several FFT images we have observed the
splitting of the circle at ${\mathbf K}_{\mathrm int}\simeq 2{\mathbf
k}_{\mathrm pl}$ (a hint on such a splitting may be found on the
Figure 4d). But the largest possible scan size in our setup is
25 $\mu$m$\times$<!-- -->25 $\mu$m and the improvement of the FFT
resolution is needed to study such effects.

#  Acknowledgments

Authors thank Y.E. Petrov for help with the silver coating of
cantilevers and prisms and A.M. Lifshits for supply of optical fibers.
The present research was supported by RFFI grant N 98-02-17206a and by
programs "Fundamental spectroscopy" and "Fundamental metrology" of
Russian Ministry of Science.

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