Multiple patterning (or multi-patterning) is a class of technologies for manufacturing integrated circuits (ICs), developed for photolithography to enhance the feature density. It is expected to be necessary for the 10 nm and 7 nm node semiconductor processes and beyond. The premise is that a single lithographic exposure may not be enough to provide sufficient resolution. Hence additional exposures would be needed, or else positioning patterns using etched feature sidewalls (using spacers) would be necessary. Even with single exposure having sufficient resolution, extra masks have been implemented for better patterning quality such as by Intel for line-cutting at its 45nm node or TSMC at its 28nm node. Even for electron-beam lithography, single exposure appears insufficient at ~10 nm half-pitch, hence requiring double patterning. Double patterning lithography was first demonstrated in 1983 by D.C. Flanders and N.N. Efremow. Since then several double patterning techniques have been developed such as self alignment double patterning (SADP) and a litho-only approach to double patterning. Pitch double-patterning was pioneered by Gurtej Singh Sandhu of Micron Technology during the 2000s, leading to the development of 30-nm class NAND flash memory. Multi-patterning has since been widely adopted by NAND flash and random-access memory manufacturers worldwide. Situations requiring multiple patterning There are a number of situations which lead to multiple patterning being required. Sub-resolution pitch The most obvious case requiring multiple patterning is when the feature pitch is below the resolution limit of the optical projection system. For a system with numerical aperture NA and wavelength λ, any pitch below 0.5 λ/NA would not be resolvable in a single wafer exposure. The resolution limit may also originate from stochastic effects, as in the case of EUV. Consequently, 20 nm linewidth still requires EUV double patterning, due to larger defectivity at larger pitches. Two-dimensional pattern rounding It is well-established that dense two-dimensional patterns, which are formed from the interference of two or three beams along one direction, as in quadrupole or QUASAR illumination, are subject to significant rounding, particularly at bends and corners. The corner rounding radius is larger than the minimum pitch (~0.7 λ/NA). This also contributes to hot spots for feature sizes of ~0.4 λ/NA or smaller. For this reason, it is advantageous to first define line patterns, then cut segments from such lines accordingly. This of course, requires additional exposures. The cut shapes themselves may also be round, which requires tight placement accuracy. Line tip vs.