1. C entre for A nalysis, S cientific computing and A pplications http://www.casa.tue.nl

2. CASA Consultancy Focus on projects at industry Small projects (vouchers) Large projects (ASML, …) Christina Giannopapa Marc Noot Ronald Rook Bas v/d Linden Martijn Slob Jurgen Tas Annemarie Aarts

3. STARS software subcontracting Optical Maskless Lithography Potential projects STARS: 3D FEM model Philips EM&C Electromagnetic signal modeling Current projects Laser drilling modeling Shape optimization Radiation modeling

4. Project: Optical Maskless Lithography 1.Overview of lithography process 2.The idea of maskless lithography 3.What is the Big Problem? 4.Mathematical formulation

5. Slicing Polishing Material deposition or modification Photoresist coating Exposure (step and scan) Developing and baking Etching and ion implantation Removing the photoresist (ashing) Completed wafer Separation Packaging ASML Other Suppliers Overview of lithography process

6. Mask based lithography Mask: glass plate with chrome pattern Optical system: 4x reduction factor Die: One layer for one integrated circuit Wavelength (λ) 193 nm (UV) Feature size on die: 50 nm ≈ 0.25 λ on mask: 200 nm Costs of mask € 1 million

7. The idea of maskless lithography “Avoid high mask investment by generating light distribution using movable mirrors”

8. The idea of maskless lithography Spatial Light Modulator (SLM) 2100 x 5200 mirrors Mirror dimension: 8μm x 8μm (on die: 20 nm x 20 nm ) (20 nm ≈ 0.10 λ)

9. The Big Problem Optical system has reduction factor of 400x → 25.000 exposures needed for 1 die 11 million mirrors in one SLM → 25.000 x 11 million = 2.7x10 11 angles need to be calculated Computing budget: 10 16 floating point operations

10. Mathematical modeling A lens is a low-pass filter Local influence of mirror Coupling with about 20x20 mirrors

11. Mathematical approaches Frequency domain Spatial domain + Easy to implement filter behavior of the optical system - Unable to make use of local influence of mirror - Filter behavior of lens must be Implemented using convolution (expensive!) + Local influence of mirrors result in sparse system. FFT

12. Mathematical challenges Equations are non-linear use some Newton method (least squares minimization: Gauss-Newton) use efficient approximations of Jacobian System is underdetermined multiple solutions exist Angles are constrained how to implement this?

13. Current approach Calculate filtered residual in frequency domain. (Use FFT to switch to/from domains) Forget about filtering in Jacobian. Jacobian becomes diagonal Trivial solution of Newton step More steps needed, but experiments show good convergence This approach would lead to a complexity of FLOPS which meets budget requirement