We present an analytical formula to predict the three-dimensional field distribution of a nanoscale bowtie aperture using quasi-spherical waves (QSWs) and surface plasmon polaritons, which are excited by the fundamental waveguide mode and local plasmons of the aperture, respectively. Assuming two separate bowtie apertures in a metal film, we analysed the decay characteristics of QSWs using a finite difference time-domain method. To verify the formula, we recorded the spot patterns of the bowtie aperture on a photoresist film using various exposure times, and fit the patterns to the analytical formula in terms of the width and depth of the patterns. In addition, it was found that the formula successfully represented the dipole characteristics of the spot patterns, which were in agreement with the surface geometry, with a root-mean-square error of 9.4%. We expect that our theoretical formula will extend the potential applications of nanoscale bowtie apertures to plasmonic device fabrication, three-dimensional plasmonic lithography, and other technologies.
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