# Rigorous Three-Dimensional Time-Domain Finite-Difference Electromagnetic Simulation

### Alfred K.-K. Wong

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/ERL M94/69

1994

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/ERL-94-69.pdf

This thesis describes the latest embodiment of a three-dimensional electromagnetic simulation program called TEMPEST which is implemented on the connection machines CM-2 and CM-5, and is used to predict and study technology trade-offs of interest in photolithography. Highlights of the new algorithm include generalization to three-dimensional calculation, analysis of dispersive materials, an efficient absorbing boundary condition, oblique incidence, and image synthesis based on Hopkins' formulation. The finite propagation speed of electromagnetic waves makes the time-domain finite-difference approach a natural choice for implementation on parallel computer architectures. This thesis addresses algorithmic issues including the accuracy and stability of the numerical scheme, and numerical boundary conditions. The conventional time-domain finite-difference scheme is second order accurate and requires 15 simulation nodes per wavelength to achieve a 2% accuracy. Stability of the scheme depends on the ratio between the spatial and temporal discretizations. Analogous to previous work in plasma physics, instability of the algorithm due to highly dispersive materials has been alleviated by calculating explicitly the time-domain convolution relation between the electric field and the electric displacement. A novel boundary condition derived from the harmonic nature of electromagnetic waves is used to bound the simulation domain with minimal artificial reflection. Implementation of a software package which caters to user convenience in data processing and remote use of the connection machine is also included. A link between TEMPEST and the optical image simulation program SPLAT allows the study of the interaction among mask topography effects, partial coherence effects, and lens aberrations. Validation of TEMPEST via experimental comparison and the usefulness of the program in predicting and assessing complex technological issues in photolithography are presented. In particular, TEMPEST is used to predict important effects such as glass edge scattering in phase-shifting masks and resonance in dielectric ridges. These predictions have been subsequently confirmed experimentally. Transmission loss and polarization effects in small contact holes are characterized as a function of the feature size. TEMPEST is also shown to be well-suited for analyzing three-dimensional effects of reflection from underlying topography during photoresist exposure which can cause variations in the critical dimensions of the features being formed.

**Advisor:** Andrew R. Neureuther

BibTeX citation:

@phdthesis{Wong:M94/69, Author = {Wong, Alfred K.-K.}, Title = {Rigorous Three-Dimensional Time-Domain Finite-Difference Electromagnetic Simulation}, School = {EECS Department, University of California, Berkeley}, Year = {1994}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/2614.html}, Number = {UCB/ERL M94/69}, Abstract = {This thesis describes the latest embodiment of a three-dimensional electromagnetic simulation program called TEMPEST which is implemented on the connection machines CM-2 and CM-5, and is used to predict and study technology trade-offs of interest in photolithography. Highlights of the new algorithm include generalization to three-dimensional calculation, analysis of dispersive materials, an efficient absorbing boundary condition, oblique incidence, and image synthesis based on Hopkins' formulation. The finite propagation speed of electromagnetic waves makes the time-domain finite-difference approach a natural choice for implementation on parallel computer architectures. This thesis addresses algorithmic issues including the accuracy and stability of the numerical scheme, and numerical boundary conditions. The conventional time-domain finite-difference scheme is second order accurate and requires 15 simulation nodes per wavelength to achieve a 2% accuracy. Stability of the scheme depends on the ratio between the spatial and temporal discretizations. Analogous to previous work in plasma physics, instability of the algorithm due to highly dispersive materials has been alleviated by calculating explicitly the time-domain convolution relation between the electric field and the electric displacement. A novel boundary condition derived from the harmonic nature of electromagnetic waves is used to bound the simulation domain with minimal artificial reflection. Implementation of a software package which caters to user convenience in data processing and remote use of the connection machine is also included. A link between TEMPEST and the optical image simulation program SPLAT allows the study of the interaction among mask topography effects, partial coherence effects, and lens aberrations. Validation of TEMPEST via experimental comparison and the usefulness of the program in predicting and assessing complex technological issues in photolithography are presented. In particular, TEMPEST is used to predict important effects such as glass edge scattering in phase-shifting masks and resonance in dielectric ridges. These predictions have been subsequently confirmed experimentally. Transmission loss and polarization effects in small contact holes are characterized as a function of the feature size. TEMPEST is also shown to be well-suited for analyzing three-dimensional effects of reflection from underlying topography during photoresist exposure which can cause variations in the critical dimensions of the features being formed.} }

EndNote citation:

%0 Thesis %A Wong, Alfred K.-K. %T Rigorous Three-Dimensional Time-Domain Finite-Difference Electromagnetic Simulation %I EECS Department, University of California, Berkeley %D 1994 %@ UCB/ERL M94/69 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/2614.html %F Wong:M94/69