Temporal Behavior of Photobleaching in Scanning Near Field Optical Microscopy
Othmar Marti, Robert Brunner, and Alexander Bietsch
Abteilung Experimentelle Physik
University of Ulm
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
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
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=2R to 1024R 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.
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.
1 D. W. Pohl, W. Denk, and M. Lanz, Appl. Phys. Lett. 44, 651 (1984).2 A. Lewis, M. Isaacson, A. Harootunian, and M. Muray, Ultramicroscopy 13, 243 (1984).3 L. Novotny, D. W. Pohl, and P. Regli, J.Opt.Soc.Am.A-Opt.Image.Sci. 11, 1768 (1994).4 E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, Science 251, 1468 (1991).5 E. Betzig and J. K. Trautman, Science 257, 189 (1992).6 A. J. Meixner, D. Zeisel, M. A. Bopp, and G. Tarrach, Opt.Eng. 34, 2324 (1995).
Please send an e-mail with questions or comments on this web site to : firstname.lastname@example.org