In the datapath, anything that can be computed using edge-based tools can also be computed using pixels (and vice versa). Information theory, specifically the Nyquist-Shannon Sampling Theorem, established the mathematical basis for digitalization in the first half of the 20th century. This math also established the basis for pixel-based computing.
Today, this math is used routinely in many fields and applications. So, while it may seem incredible that any smooth curve could be represented by pixels, we all know through everyday experience that this is true, because we see it in such things as television, LCD displays, and computer games. In semiconductor manufacturing, we see it in multi-beam mask writing.
D2S uses this same math to compute in the pixel domain. Pixel-domain solutions offer the ability to manipulate any shape, including conventional or complex Manhattan shapes, or diagonals, or any curvy shapes with the same accuracy and runtime. D2S presented a paper at SPIE Photomask Technology 2022 (awarded Best Paper) that confirms pixel-based computing also offers a very high degree of accuracy.
Abhishek Shendre, Aki Fujimura, Mariusz Niewczas, Tom Kronmiller, “You don’t need 1nm contours for curvilinear shapes: pixel-based computing is the answer,” Proc. SPIE 12293, Photomask Technology 2022, 1229307 (1 December 2022); doi: 10.1117/12.2643339
O(p) = Shape/Complexity Agnostic
With pixel-based computing, runtime is constant (not just linear, but constant) over the number of pixels. The number of pixels needed is fixed for any given technology node based on the resolution available from a given lithography technology or multi-beam writer. With CPU-centric computing based on polygon edges, runtime for any operation, O, is O(n*log(n)) or O(n^1.4), where n is the number of polygon edges. So as shapes get more complex with more polygon edges, algorithms run slower and slower. Pixel-based computing is O(p) where p is a constant for a given technology node.
Because D2S solutions are in the O(p) domain, they are shape agnostic. Manhattan and curvilinear shapes – even with the full reticle being Entirely Manufacturable™, and therefore entirely curvilinear – are processed in the same amount of time with the same uniformity and linearity benefits and the same hardware and software cost. The entry price and cost of operations is the same, no matter the complexity.
The shift to the pixel domain for masks naturally leads to the use of GPU-accelerated computing, which excels in pixel-domain computation. The GPU-accelerated D2S computational design platform (CDP) was a major enabler for both the D2S TrueMask® ILT breakthrough for Entirely Manufacturable ILT masks and for D2S pixel-level dose correction (PLDC), which performs mask process correction (MPC) either inline, at the same time as the machine writes the mask, or offline as part of the ILT flow before handoff to mask writing.
