4.4 Boolean Operations

If geometries are represented as level sets, Boolean operations can be expressed as simple operations on the corresponding LS functions [89,110]. If one considers the sets ${\mathcal{M}}_A$ , ${\mathcal{M}}_B$ , and ${\mathcal{M}}_C$ and the corresponding boundaries ${\mathcal{S}}_A=\partial{\mathcal{M}}_A$ , ${\mathcal{S}}_B=\partial{\mathcal{M}}_B$ , and ${\mathcal{S}}_C=\partial{\mathcal{M}}_C$ , which are represented by the LS functions ${\Phi}_A$ , ${\Phi}_B$ , and ${\Phi}_C$ , the LS counterparts of Boolean operations are listed in the following.

$${\mathcal{M}}_A ={\mathcal{M}}_B \cup {\mathcal{M}}_C \Leftrightarrow {\Phi}_A =\min({\Phi}_B,{\Phi}_C)$$ Union: (4.10)

$${\mathcal{M}}_A ={\mathcal{M}}_B \cap {\mathcal{M}}_C \Leftrightarrow {\Phi}_A =\max({\Phi}_B,{\Phi}_C)$$ Intersection: (4.11)

$${\mathcal{M}}_A =\mathbb{R}^{D}\setminus{\mathcal{M}}_B \Leftrightarrow {\Phi}_A =-{\Phi}_B$$ Complement: (4.12)

$${\mathcal{M}}_A ={\mathcal{M}}_B\setminus {\mathcal{M}}_C \Leftrightarrow {\Phi}_A =\max({\Phi}_B,-{\Phi}_C)$$ Relative complement: (4.13)

Here the convention is used that the LS function ${\Phi}_X$ is negative for all ${\vec{x}}\in{\mathcal{M}}_X$ .

Boolean operations are very useful for more general topography simulations, if consecutive process steps, like etching and/or deposition processes, should be simulated or several materials are involved.