• Finding optimal photomask layouts and illumination settings.
  • Off-axis illumination, optical proximity correction (OPC).
  • No user intervention, very few preconditions.
  • Mutual optimization for different features (e.g., chained/isolated contact hole set-up)
  • Underlying optimizer: genetic algorithm (GA).
  • Related Work: Rosenbluth et al., Smith, Oh et al., Socha et al., Granik, and Poonawalaa et al.

Genetic Algorithm

  • Global optimization method.
  • Non-analytical technique → requires little a priori
  • knowledge of the search space.
  • Easily adaptable to different tasks.
  • Mimics “natural” evolution or, more precisely, breeding: selection, recombination, and mutation.



GA Flow/Simulator Interface

optional local optimizer, e.g., via LisBON
Christoph Dürr (2006)

Parameter Representation

  • GA uses binary “alphabet” → binary encoding of parameters. E.g., bin(5) = (101)2.
  • Mask elements are restricted to rectangles → 2 vertices/4 parameters per element.
  • Illumination subdivided in track/sector pixels.
  • Each pixel can take transmission values of 0 or 1.
  • Transformation into simulator’s representation (mesh grid).

  • Up to 50 parameters (mutual optimization of two features).

Mask Representation

Mask Representation


  • Merit of a solution is determined by aerial image computation and evaluation (Kirchhoff approach).
  • Aerial image analysis is conducted at different de-focus positions (e.g, 0 – 400 nm).
  • In a first step, aerial images are computed at low resolution (10 mesh-points).
  • If obtained merit exceeds a pre-defined threshold, a finer grained (20 mesh-points) evaluation is conducted.

Mask Manufacturability Criterion

In order to obtain a producible mask, following conditions should be met:

  1. Elements should exhibit minimum dimensions and
  2. a minimum distance between elements should be


CD Criterion

  • Obtained CDs (critical dimensions) should match the target CD as closely as possible.
  • For each feature and each defocus setting, target and computed CDs are compared.
  • The difference is considered the CD badness.
  • The overall merit is decreased

Band Criterion

  • Prevention of side-lobe-printing.
  • Assurance of main-feature-printing.
  • “Security Band” around threshold: both side-lobes and main features hitting this band decrease the overall merit.

Merit Function



  • NA := 0.7,
  • λ := 193 nm,
  • High transmission (19.3 %) attenuated PSM
  • Objective: Finding optimum mask and illumination set-ups for 140 nm x 170 nm contact holes
  • pitches:
    dense set-up dense: 320 nm x 410 nm,
    chain set-up chain: x-pitch: 320 nm,
    chain set-up isolated