Optimization of real phase-mask performance

TLDR: In this paper, the Bessel function was applied to the phase mask design and the results showed that the image was significantly brighter than with other techniques, and the depth of focus was also significantly increased.

Abstract: Classical phase mask lithography designs can be understood by the application of Fourier optics to the phase mask pattern. For maximum resolution, the mask design for a circularly symmetric contact hole will have a Fourier pattern with most of the energy near the edge of the system aperture. The inverse Fourier transform of an ideal annular Fourier plane pattern is the Bessel function, Jo. This function consists of a central lobe and an infinite number of rings, with each ring having equal energy and alternating phase shifts. This function is a solution to the wave equation in cylindrical coordinates. A characteristic of this solution is that the field has the same transverse profile, independent of the position along the axis. Therefore, the diameter of the central lobe of an ideal Bessel beam will have infinite depth of focus. Although approximations to such diffraction free beams have been reported for coherent light, the depth of focus depends on the number of outer lobes of the Bessel function that can be produced. Practical mask design and imperfect coherence limit the number of lobes that are actually useful. However, a coarse approximation to the Bessel beam can be created with only one or two phase shifted lobes. This is what is attempted in the optimized "outrigger" patterns. The ideal design, however, is one in which one phase- edge is placed at the zero of the Jo function, and a second is placed so that the power in the outer lobe approximates that of the Bessel function. Masks were fabricated with such designs, and the wafer exposures show that the image is significantly brighter than with other techniques, and the depth of focus is also significantly increased.