It was recently proposed to put fluorescent molecules at the end of the tip of a scanning near-field optical microscope (SNOM). By measuring lifetime modifications during the tip scanning, one can obtain an image of the sample. We deal with a model of this lifetime SNOM. We propose a theoretical study of the decay rate or the level width of a fluorescent molecule located above a rough interface. When the surface relief is small compared with the wavelength, we determine the electromagnetic decay rate by using a first-order approximation of the Rayleigh–Fano method. Within this approximation the electromagnetic decay rate is composed of two terms: a specular term, which depends only on the distance from the molecule to the sample mean plane, and a diffraction term, which depends on the coordinates of the molecule and on the Fourier transform of the interface profile. It is demonstrated that a kind of linear transfer function can be introduced that connects the Fourier transform of the interface profile to the Fourier transform of the diffraction decay rate. It is independent of the profile but depends on the dielectric constants, on the molecular dipole orientation, and on the molecule–surface distance. We study the properties of this transfer function for a dielectric interface and for various molecular dipole orientations and then present some calculated images. The existence of this transfer function and the good visibility of the calculated images are positive indications for the use of this new kind of SNOM.
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