Modeling of Discharge Stability in High-Voltage Power Supplies for Excimer Lasers

Excimer lasers, as critical light sources for semiconductor lithography and medical applications, rely on precise control of high-voltage power supply discharge for stable output. Discharge stability modeling involves multiphysics coupling of gas dynamics, circuit response, timing synchronization, and thermal effects, representing a core technological direction for enhancing laser performance. 
1. Gas Dynamics and Discharge Stability Modeling
The discharge stability of excimer lasers is primarily influenced by the properties of the working gas. For example, in ArF lasers (193 nm), the buffer gas type directly determines electron density distribution and pre-ionization effects. Studies show: 
• When neon (Ne) serves as the buffer gas, the electron depletion layer width (~7 μm) and sheath layer width (~11 μm) near the cathode are significantly smaller than in helium (He) systems (15 μm and 20 μm, respectively). This is due to Ne’s stepwise ionization and secondary ionization processes, which replenish free electrons and suppress local electric field distortion. 
• Adding trace xenon (Xe) further optimizes discharge stability. Xe’s ionization energy (12.1 eV) is lower than the photon energy (14.6 eV) emitted by Ne’s excited states. Through photoionization (Xe + hν → Xe⁺ + e⁻), initial electron density increases, reducing the discharge threshold voltage by ~15% and accelerating discharge region expansion. 
A hydrodynamic model quantifies these processes: By solving the Boltzmann equation for electron collision reactions, combined with particle continuity equations and self-consistent electric field equations, the transient evolution of inter-electrode voltage, current, and photon number density is simulated, predicting discharge cycles (~120 ns for Ne) and optical pulse durations (~25 ns for Ne). 
2. Dual-Cavity Synchronization and Pulse Timing Control Model
High-repetition-rate dual-cavity excimer lasers (e.g., Master Oscillator MO and Power Amplifier PA) require strict synchronization (jitter < ±5 ns); otherwise, seed light amplification efficiency decreases. Synchronization control addresses four jitter sources: 
• Power supply starting voltage jitter: High-precision DC power supplies (error < 1‰) charge both cavities simultaneously. 
• Timing drift: Impedance matching circuits and constant-fraction timing chips measure the actual light emission delay between MO/PA. Closed-loop control algorithms (e.g., proportional-integral) dynamically adjust trigger pulses. For instance, a delay output unit uses a programmable module (resolution 0.25 ns) and fixed delay lines (compensating inherent errors of ±200 ns) to generate two optical pulse signals, compressing synchronization jitter to within ±5 ns. 
• Temperature and pressure drift: A state acquisition unit monitors cavity temperature and pressure in real time, feeding back to adjust discharge voltage amplitude and phase. 
3. Multi-Objective Optimization Control Model
To balance single-pulse energy stability and dose accuracy (e.g., lithography dose accumulation requirements), a multi-objective optimization model is established: 
• Segmented energy control: Pulse sequences are divided into overshoot (i<g) and non-overshoot (i>g) segments. The overshoot segment uses a PI algorithm to calculate discharge high voltage: 
  \[
  \text{HV}_{E_{m+1,i}} = K_{E_o} \cdot (E_{m,i} E_t) + I_{E_o} \cdot \sum (E_{m,i} E_t)
  \] 
  The non-overshoot segment introduces cumulative error terms from previous pulses to suppress low-frequency drift. 
• Thermal effect constraints: Discharge energy E \propto U \cdot I \cdot t causes temperature rise, necessitating a thermal resistance term in the objective function: 
  \[
  F = w_1 \cdot \text{HV}_{E_{m,i}} + w_2 \cdot \text{HV}_{D_{m,i}} w_3 \cdot \text{HV}_{T_e}
  \] 
  where \text{HV}_{T_e} = R_{\text{th}} \cdot \frac{U^2}{R} (R_{\text{th}} is thermal resistance), with weighting coefficients w_1 + w_2 + w_3 = 1. 
• Genetic algorithm solution: Voltage regulation range, slope efficiency, and response speed are treated as constraints. Chromosome encoding and fitness function P = \tau \cdot F iteratively solve for the optimal discharge voltage setting, balancing energy stability and thermal management. 
4. Experimental Validation and Industrial Significance
Simulation platforms based on these models predict discharge stability under different buffer gas ratios and power supply parameters. For example, Ne-Xe mixtures increase discharge threshold current by 2.1× (from 0.85 mA to 1.78 mA), and airflow cooling (>102 m/s) suppresses anode bright spot formation, expanding the glow discharge region. This modeling approach has been applied to high-repetition-rate lasers (4 kHz), achieving single-pulse energy fluctuation < ±1% and dose accuracy >99%, providing theoretical support for long-term operation of semiconductor lithography light sources.