**Temporal Behavior of Photobleaching in Scanning Near Field Optical Microscopy**

Othmar Marti, Robert Brunner, and Alexander Bietsch

*Abteilung Experimentelle Physik*

*University of Ulm*

*D-89069 Ulm*

*Germany*

The measured fluorescences signal of a quasi-twodimensional layer of dye molecules
iradiated through a sub-wavelength aperture of a scanning near filed optical microscope
exhibits a logarithmic time dependence over more than two decades in time. The nonuniform
illumination by a point like source bleaches the center of the irradiated area much faster
than the outer parts. Hence the bleaching of the fluorescence signal is governed by a
distribution of exponential decays with position dependent time constants. A simple model
taking into account the size of the aperture and the diameter of the detection area
reproduces the measured data. It is deduced that the time scale over which a logarithmic
decay is observed is a direct measure of the probe size and the size of the detection
area. Hence we propose to use a measurement of the time evolution of the fluorescence
signal to determine the resolution in scanning near field optical microscopy.

Optical measurement techniques have become invaluable tools in semiconductor physics,
polymer physics and in the research on biological samples. Since the sizes of the observed
objects often is below the wavelength of the light the scanning near-field optical
microscope (SNOM) was developed1. A probe with a small, sub wavelength aperture2 for the
light is raster scanned across the surface. Theoretical calculations3 have demonstrated
that the diameter of the aperture determines the resolution. This aperture is often
fabricated from a tapered fiber4 covered by a thin metal film. SNOMs habe been operated in
transmission or reflection mode5. Often spectras, polarizations and the time dependences
are determined in addition.

Recently it became popular to do research on single molecules, where the fluorescence
signal is used to

fluorescence

bleaching

We are interested in the time dependence of the bleaching of the fluorescence in a thin
layer of molecules. This monomolecular layer is assumed to be two-dimensional. To
calculate the temporal response in the Scanning Near Field Optical Microscope we start
with the rate of fluorescence for a single molecule

where s is the absorption cross section of the molecule, *I(r)* the excitation
intensity and the energy of one photon6. is the fluorescence quantum yield. The number of
fluorescence photons produced below the tip can be written as

where the area density of fluorescent molecules n(r,t) is a function of position and
time. The integration is over the area which is mapped onto the photodetector. Every
molecule bleaches out, on the average, after undergoing a characteristic number of
fluorescence cycles. We can therefore formulate the bleaching rate for one molecule with
the rate

where is the quantum yield for bleaching. The number of molecules obeys then the law .
We can integrate the previous equation and obtain for the number of molecules

Equations - are generally valid for every intensity distribution conceivable in the a
scanning near-field optical microscope. We will first solve the equation for a point like
light source. Later we will generalize the solution for an illuminated disk like aperture.
If we label the distance between the point source and the sample surface by *R* and
if we take into consideration that the intensity in a spherical wave is inversely
proportional to the square of the distance to the source we obtain

where *P0* is the laser power emitted from the point source through the aperture
the factor 4p results from the fact that the integral over a sphere enclosing the source
should recover the Power *P0*. Substituting equation into equations - gives the
result

where we have carried out the integration over the angle . *rmax* is the radius of
the illuminated area, or altenatively, the radius of the area projected on the
photodetector. If we start the experiment with an uniform distribution of molecules we can
set . Therefore we can use the substitutions and we obtain

The solution of the integral in equation is composed of two exponential-integral
functions

where

For large times the two functions tend to 0, whereas for small times their difference
aproaches a constant value which is only a function of *rres*. In the intermediate
range the second summand in is dominant. In this same time range the sum in equation is
negligible compared to the logarithm. Since the logarithm does not depend on the prefactor
of the exponential, one arrives to the conclusion that in a certain time range one can
write

From equation we can conclude that in a limited range the fluorescence intensity should
follow a logarithmic law, independent of the factor in equation . Fig. 1 shows some curves
calculated in reduced coordinates with *R*=1, *A*=1, *b*=*1.* The time
range where the logarithmic bleaching law is valid depends on the integration area. The
larger the integration area is the more orders of magnitudes in time exhibit the linear
law. The lines are calculateed for integration radii of *rres*=2*R* to 1024*R*
in powers of two. The smallest integration area corresponds to the lowest curve. All
curves for the larger radii have the same slope, which is to be expected from equation .

As a comparison we have included the measured fluorescence time dependence of a ...
monolayer on glass measured with a near field optical microscope. To adjust the slope of
the measured data we find a scale factor . The second adjustment is the time scale of the
measurement. We find that . There is no possibility to fit the data to curves with an
integration area with a diameter smaller than 128R. The adjustment of the slope *A*
sets the magnitude of the fluorescence rate at small times. If the measured fluorescence
rate for one set of apertures exceeds the thearetically possible values at small times
then this aperture is too small. Hence the simple theory outlined above allows to give a
lower limit of the illuminated area, unlike other measurements of the resolution: these
usually give an upper limit of the imaged area.

From the parameters A and b we can calculate the relationship of the bleaching quantum
efficiency and the fluorescence quantum efficiency

**Figures**

Fig. 1 Calculated time dependence of the intensity as a function of the integration
range. Plotted is the fluorescence intensity versus the log10 of the time. The calculated
lines are for *rmax* = 2 to 1024, in powers of 2, from the bottom to the top. The
points are measured data.

**Fig. 1**

**References**