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Optimal decentralized wavelength control in light sources for lithography

Mruganka Kashyap

Abstract

Pulsed light sources are a critical component of modern lithography, with fine light beam wavelength control paramount for wafer etching accuracy. We study optimal wavelength control by casting it as a decentralized linear quadratic Gaussian (LQG) problem in presence of time-delays. In particular, we consider the multi-optics module (optics and actuators) used for generating the requisite wavelength in light sources as cooperatively interacting systems defined over a directed acyclic graph (DAG). We show that any measurement and other continuous time-delays can be exactly compensated, and the resulting optimal controller implementation at the individual optics-level outperforms any existing wavelength control techniques.

Optimal decentralized wavelength control in light sources for lithography

Abstract

Pulsed light sources are a critical component of modern lithography, with fine light beam wavelength control paramount for wafer etching accuracy. We study optimal wavelength control by casting it as a decentralized linear quadratic Gaussian (LQG) problem in presence of time-delays. In particular, we consider the multi-optics module (optics and actuators) used for generating the requisite wavelength in light sources as cooperatively interacting systems defined over a directed acyclic graph (DAG). We show that any measurement and other continuous time-delays can be exactly compensated, and the resulting optimal controller implementation at the individual optics-level outperforms any existing wavelength control techniques.
Paper Structure (14 sections, 1 theorem, 12 equations, 6 figures)

This paper contains 14 sections, 1 theorem, 12 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Decentralized LQG control
  3. Problem Setup
  4. Cost
  5. Time-delays and DAG
  6. Decentralized wavelength control
  7. Solution outline
  8. Continuous-time sub-system level controller
  9. Discussion
  10. Finite horizon cost
  11. Discrete-time decentralized control
  12. Optimality of performance
  13. Aliased disturbances
  14. Conclusions

Key Result

Lemma 1

kashyap2023thesis Consider the global plant $\mathcal{P}$ defined as $\left(\right)=\left[\right] \left(\right)$, which is formed by stacking the dynamics of $N$ sub-systems, where the regulated output $z=\left(\right)^\tp x + \left(\right)^\tp u$ for cost matrices Q and R. Then the optimal $\mathca

Figures (6)

  • Figure 1: Schematic of a Line Narrowing Module (LNM) based on multi-prism dispersion theory. An LNM is an example of a combination of optics used for wavelength and bandwidth control in light sources.
  • Figure 2: Example of a 2-node directed acyclic graph (DAG). The communication graph has the structure of $\mathcal{S}$, shown on the right. The global matrices $A$ and $B_2$ belong to the set $\mathcal{S}$.
  • Figure 3: A representation of a standard prism-actuator combination in modern-day light sources used in lithography. The system is in continuous-time, while the controller is digital, requiring sample (LAM) and hold blocks. Note that we are dealing with a continuous-time system with a discrete-time controller.
  • Figure 4: The 2-Prisms (with their actuators) and corresponding controllers are abstracted as 2-node directed acyclic graph (DAG). Without considering any delays, the communication graph has the structure $\mathcal{S}=\left(****\right)$.
  • Figure 5: The continuous-time controller for the PZT $+$ Prism $3$ is a combination of a Kalman filter (red box with gain $L_P$), a regulator (blue box with gain $\tilde{F}_P$) and two FIR blocks: $\Pi_{u_P}$ and $\Pi_{b_P}$. A correction control signal $v_{PS}$ is transmitted to the controller for Stepper $+$ Prism $4$, while a similar delayed information is received from the Stepper combination. Here $\mathcal{T}\defeq \left(sI-\left(A_{P}00A_{S}\right)-\left(L_{P}C_{2_P}000\right)\right)^{-1}$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Lemma 1